Ekström was active in high-level national Swedish and international chess during a short period of about five years in the mid-1940s, with some very impressive successes. He won at Stockholm 1942, tied with Stig Lundholm, ahead of both Gösta Stoltz and Erik Lundin, both of whom became Grandmasters later on. Then at Stockholm 1943–44, he won ahead of Stoltz. Ekström lost a 1944 match to the world-class Grandmaster Paul Keres by 5–1, following Keres' 'hors concours' appearance at the 1944 Swedish Championship, where he had placed second.
Ekström then finished second himself at the strong Hastings 1945–46 tournament, just half a point behind Grandmaster Savielly Tartakower, with a score of 9/11. This was ahead of former World Champion Max Euwe, American champion Arnold Denker, and American Olympian Herman Steiner, who all trailed well behind with 7 points. At Zaandam 1946, Ekström shared second with László Szabó, an eventual nine-time Hungarian champion and three time Candidate, with 8.5/11. The winner was Euwe, who made 9.5/11. Swedish Olympian Stoltz was next with 8. Then, at Stockholm 1946–47, Ekström tied for first with Lundin. They scored 7/9, ahead of (among others) Swedish Olympian Gösta Danielsson and Finnish champion Eero Böök, who shared third with 6.
Other than a couple of minor Swedish team events, this seems to be the recorded extent of Ekström's competitive chess career in over-the-board play. He never represented Sweden in Chess Olympiad competition, although he very well could have, based upon his successes, as he was finishing ahead of team members in tournaments.
The website chessmetrics.com, which endeavours to rate historical chess performances and players from the times before International chess ratings were established in 1970, puts Ekström at #9 in the world, with a rating of 2714, in January 1947. This is in the super-grandmaster range. The chessmetrics data does not incorporate his two Stockholm successes of 1942 and 1943–44. This estimate, which is based upon less data than the chessmetrics ratings of other top players of that time, may be a bit on the generous side, considering his one-sided 1944 loss to Keres, who was an undoubted top five player.
Ekström was awarded the International Master title by FIDE, the World Chess Federation, in 1950. He played correspondence chess with success as well, earning the IMC title in 1971.
The Ekström Variation of the Queen's Gambit Declined is named for him. It runs 1.d4 d5 2.c4 e6 3.Nc3 Nf6 4.Nf3 c6 5.Bg5 dxc4 6.e4 b5 7.e5 h6 8.Bh4 g5 9.exf6 gxh4 10.Ne5.
A selection of 65 of Ekström's games can be found at chessbase.com.