Sangakkara hit a century against Bangladesh on the first day of the first Test at Galle International Stadium and moved ahead of Gavaskar and into 10th position in all-time leading run scorers in Test cricket.
The left-hander went past Gavaskar's 10,122 runs and is currently on 10,187 after scoring 142, his 31st Test hundred. Sangakkara is playing his 116th Test.
Mahela Jayawardene is the only other Sri Lanka batsman to have crossed 10,000 runs in five-day cricket.
For the first time after more than 33 years, Gavaskar has been displaced from the top 10 of leading run scorers. The former Indian opener was the first to reach the 10,000-run milestone.
Top 10 run scorers in Test cricket (Read as player, country, matches, runs)
Sachin Tendulkar (India) 196, 15,746
Ricky Ponting (Australia) 168, 13,378
Rahul Dravid (India) 164, 13,288
Jacques Kallis (South Africa) 162, 13,128
Brian Lara (West Indies) 131, 11,953
Allan Border (Australia) 156, 11,174
Steve Waugh (Australia) 168, 10,927
Mahela Jayawardene (Sri Lanka) 138, 10,806
Shivnarine Chanderpaul (West Indies) 146, 10,696
Kumar Sangakkara (Sri Lanka) 116, 10,187