Volkov turned professional in 1988. His first noteworthy result came in 1990, when he stunned the tennis world by defeating Stefan Edberg (who was ranked the World No. 1 at the time) in straight sets in the first round of the US Open.
After finishing runner-up in three tourmaments over 1989 and 1990, Volkov won his first top-level professional singles title in 1991 at Milan. At Wimbledon that year, he famously lost a close five set match in the fourth round to the eventual tournament champion Michael Stich, 6–4 3–6 5–7 6–1 5–7, despite winning the same number of games as Stich overall in the match, which hinged on a lucky shot struck by the German when he was trailing 3-5 in the final set. As Volkov served for the match, with the score at 30-15 in his favour, a shot struck by Stich was heading out, but the ball hit the inside edge of the top of the net post, sailed over Volkov's head, and landed inside the line, sparing the German from having to face two successive match-points on Volkov's service game, and bringing the scoreline instead to 30-30. Stich went on to take the game; and subsequently Volkov would not win another.
Volkov was runner-up in three tournaments in 1992, and won his second title in 1993 in Auckland. Later in 1993 Volkov defeated Björn Borg in the first round of the Kremlin Cup in Moscow, in a match which proved to be the last of Borg's career. Volkov won the Kremlin Cup a year later in 1994 to claim his third (and final) career title. (He reached one more final in 1997 in Shanghai.)
Volkov was part of the Russian team which reached the final of the Davis Cup in 1994. He won important singles rubbers over Patrick Rafter of Australia in the first round, and Michael Stich of Germany in the semi-finals. However he lost both his singles rubbers in the final as Russia were defeated 4–1 by Sweden.
Volkov retired from the professional tour in 1998. His career-high singles ranking was World No. 14 in 1994. His career prize-money earnings totalled US$3,362,786.
He used to be Marat Safin's coach before the duo split in July 2007.