The 35-year-old reached the milestone in his 105th Test when he trapped tail-ender Steve Harmison leg-before in the afternoon session.
Australian leg-spinner Shane Warne heads the list with 659 wickets, followed by Sri Lankan off-spinner Muttiah Muralitharan (600), Australian fast bowler Glenn McGrath (542) and retired West Indies paceman Courtney Walsh (519).
Kumble, who made his debut against England at Manchester in 1990, has grabbed five or more wickets in a Test innings 32 times and 10 or more in a match eight times.
His best came in the second Test against Pakistan at Delhi in 1999 when he finished with 10-74 to become only the second bowler after England off-spinner Jim Laker to take all 10 wickets in an innings.
Kumble, a regular feature in one-dayers before being dumped last year, has so far taken 329 wickets in 264 matches in the shorter version of the game. His best is 6-12 against the West Indies at Calcutta in 1993.