The prolific 37-year-old left-hander reached the milestone when he knocked Trent Boult square for two in Sri Lanka's first innings of the second Test against New Zealand in Wellington.
Sangakkara started the two-Test New Zealand series on 11,988 runs and after scoring six and one in the first Test in Christchurch, he was determined to get the final five in Wellington.
"When he fails, it's a coach's nightmare. You have to throw (balls to Sangakkara) for hours and hours," captain Angelo Mathews said ahead of the Test.
Since launching his Test career in 2000, Sangakkara has made an impressive average of over 58.00, including 37 centuries in 130 Tests. His highest score was 319 against Bangladesh.
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He is now number four on the all-time scoring list, although his average is higher even than Indian great Sachin Tendulkar, who tops the ranking with 15,921 runs in 200 Tests at 53.78.
Australian Ricky Ponting (13,378 at 51.85), Jacques Kallis from South Africa (13,289 at 55.37) and India's Rahul Dravid (13,288 at 52.31) round out the top five.
The ODI list is also headed by Tendulkar, who made 18,426 runs in 463 matches, ahead of Ponting with 13,704 runs from 375 ODIs and Sanath Jayasuriya of Sri Lanka, who has 13,430 runs in 445 games.