Organized chess started up again as the Second World War ended. In an All-Union Tournament of First Category players at Gorky 1945, Furman posted his first noteworthy result when he tied for first with Konstantin Klaman, at 11/15. At Tula 1945, Furman placed second with 10.5/14, behind only V. Ljublinsky. In the Leningrad Championship of 1946, Furman tied for 8th-9th places, with 8.5/17. In the USSR Championship semi-final (URS-ch15 sf), Leningrad 1946, Furman was unsuccessful in advancing, but made a highly respectable score of 9/18, to tie for 9th-10th places. He was moving up slowly through the incredibly deep Soviet vanguard.
The year 1947 brought some rewards for Furman. He tied for first place in the All-Union Championship of the Spartak Club, with Vladimir Simagin, at 15/19, but lost the playoff match. Then, in the Leningrad Championship, he tied for 3rd-4th places, with 11/17. At the Saratov 1947 National Tournament, he scored 7/11 for a tied 2nd-3rd place.
Furman qualified from the semi-final at Sverdlovsk 1947, for his first Soviet Chess Championship at age 27. In the final, he performed exceptionally well, placing third, only half a point behind the joint winners David Bronstein and Alexander Kotov, with a fine score of 11/18 (URS-ch16, Moscow 1948). In the Leningrad Championship of 1948, he tied for 7th-10th places, with 9.5/17. He tied for first-third places at Vilnius 1949, the semi-final for URS-ch17, with 11.5/17, qualifying again for the Soviet final. In the Leningrad Championship of 1949, he was off form with 8.5/18 to tie for 11th-13th places. Then, in the Soviet final later in 1949, again in Moscow (URS-ch17), he tied for 5th-7th places with 11.5/19. In the 1950 Championship of the Spartak Club, he tied for 4th-5th places, with 6/11. Then at Gorky 1950, he was unsuccessful in qualifying for the next Soviet Championship final, as he could only score 9.5/15, for fourth place.
In the URS-ch21 at Kiev 1954, Furman scored 10/19 to tie for 7th-9th places. He earned his first international tournament opportunity for Bucharest 1954, where he tied for 6th-7th places with a fine 10/17. He was in the middle of the field in URS-ch22 at Moscow 1955, with 10/19, in a tie for 10th-11th places. It was a similar story for URS-ch24, Moscow 1957, where he scored 10/21 for 12th place. He had a good tournament at Kiev 1957, scoring 11.5/19 to tie for 2nd-5th places, behind only Tigran Petrosian. His form dropped for URS-ch25, Riga 1958, as he could only make 6/18 for 17th place. At URS-ch26, Tbilisi 1959, he was again below 50 per cent with 8/19 for 15th place.
Furman gradually proved he belonged in the upper ranges of the extraordinarily deep Soviet chess elite, with many victories over top players. He placed equal fourth in the 1965 Soviet Championship. Furman was awarded the International Master (IM) title in 1954. He won the Leningrad Championship in 1953, 1954, and 1957 (jointly).
Bronstein faced Furman in the 1948 Soviet Championship, won the game, but was impressed with Furman's skill. Bronstein wrote, in his acclaimed book The Sorcerer's Apprentice (page 102): "Later, when I recognised the logical play of Furman, I invited him to be my assistant during preparation for the match [against World Champion Mikhail Botvinnik] in 1951. Also, I took him abroad as my second to the Interzonal Tournament in Gothenburg in 1955 and the Candidates' Tournament in Amsterdam 1956."
Furman had been one of the assistants to Botvinnik in his 1963 world title match against Tigran Petrosian, according to Anatoly Karpov, writing in his autobiography Karpov on Karpov. Botvinnik played many training games with Furman, to prepare for Botvinnik's 1960 and 1961 World Championship matches; these matches only became public many years afterwards, when Botvinnik published the games, which are now available on various databases, including chessgames.com. Furman and Botvinnik at that time were both members of the Trud (Trade Unions) Club. Later Furman switched to the Army Sports Club. Furman also assisted world-class players such as Viktor Korchnoi (according to both Bronstein and Karpov) and Efim Geller (according to Karpov).
The site www.chessmetrics.com, the chess site which endeavors to provide historical ratings for players while correcting for different methods of calculation, puts Furman's peak rating at 2708 in April, 1948, #11 in the world at that time. That is certain Grandmaster level, but, because of lack of international chances, Furman did not formally receive the title until 18 years later. According to chessmetrics, his best tournament from a performance rating standpoint was Gorky 1954 (5.5/6, for a 2755 performance).
From this stage on, Furman worked ever more closely with Karpov, who moved from Moscow to Leningrad, switching universities as well, from Moscow State University to Leningrad State University, to be nearer to Furman. The two also became close friends, with Karpov actually assisting Furman, upon his request, for the 1969 Soviet Chess Championship, held in Moscow; Karpov had not qualified to play in it. Karpov earned the Grandmaster title at Caracas 1970. He qualified for the Soviet Championship for the first time in 1970, scoring well. Karpov drew a 1971 secret training match with Korchnoi, a world title Candidate. His rise continued, with wins at the very strong Moscow 1971, Hastings 1971-72, and San Antonio 1972 tournaments. Karpov made the Soviet national team for the 1972 Skopje Chess Olympiad as first reserve, and scored 13.5/16, winning gold on his board, and helping the USSR to win the team gold.
Because Furman had earlier worked with Korchnoi, Karpov was able to utilize this situation to good effect to win his vital 1974 World Chess Championship Candidates final match against Korchnoi, who had earlier fallen out with Furman over a dispute around a 1971 match against Geller. Korchnoi, a difficult character, had wanted Furman to assist him against Geller, but Furman and Geller were teammates at the Army Club, so Furman withdrew his training services on principle; as Furman had also helped to train Geller. This made Korchnoi bitter against Furman, and their connection ended, even though Korchnoi went on to defeat Geller. So, a full-time training spot opened up with Furman, which Karpov took. Karpov wrote that Korchnoi did not realize at the time the strength of Karpov's potential challenge to him.
As Karpov built further successes, and was earning favour in the Soviet sports bureaucracy, he was able to arrange for Furman to compete with him sometimes in the same international tournaments, such as Madrid 1973, Ljubljana/Portorož 1975, and Bad Lauterberg 1977, all of which Karpov won. Furman also performed well, taking or tying for third place at all three of Madrid, Ljubljana / Portoroz, and Bad Lauterberg.
Furman was awarded the Honoured Trainer of the USSR in 1973 for his work with young players. He served as trainer to the combined Soviet teams to the 1974 Nice Olympiad and the 1977 European Team Championship in Moscow.
In his autobiographical book, Karpov on Karpov, published in 1991, Karpov credits Furman very extensively and deservedly with helping him scale the heights of grandmaster chess, culminating in his World Championship in 1975 and superb play for the next decade, as he dominated the game. This fine book is perhaps the most detailed work ever published on the relationship between a top chess player and his coach. The two also played a great deal of bridge together; this game became for a time something of an obsession with Furman.
Bronstein, who had worked with Furman earlier, wrote "When Furman started to work with Anatoly Karpov, I was not surprised by the young grandmaster's success, showing a brilliant understanding of grandmaster strategy. It was obvious that Furman had passed on to him a lot of the knowledge acquired during his earlier years. It should also be said that Furman had very good analytical powers and was able to look deeply into the games of other grandmasters, disclosing the secrets of their success."
Furman's health had not been good since the mid 1960s, however. He had survived one operation for stomach cancer, but the cancer returned, and he died at Leningrad in 1978, just before Karpov's match with Korchnoi for the World Championship. Karpov wrote that he missed Furman's help greatly during that match, which he won only narrowly.