The 38-year-old reached the landmark in his 319th match when he took a single off Shoaib Malik to move to 55, before being run out on 63.
Dilshan now has 10,008 runs, joining compatriots Kumar Sangakkara, Sanath Jayasuriya and Mahela Jayawardene in a coveted list headed by Indian great Sachin Tendulkar's tally of 18,426 ODI runs.
The 10,000-run club in one-day cricket (name, country, matches, runs, centuries):
Sachin Tendulkar (IND) 463 18,426 49
Kumar Sangakkara (SRI) 404 14,234 25
Ricky Ponting (AUS) 375 13,704, 30
Sanath Jayasuriya (SRI) 445 13,430 28
Mahela Jayawardene (SRI) 448 12,650 19
Inzamam-ul-Haq (PAK) 378 11,739 10
Jacques Kallis (RSA) 328 11,579 17
Saurav Ganguly (IND) 311 11,363 22
Rahul Dravid (IND) 344 10,889 12
Brian Lara (WIS) 299 10,405 19
Tillakaratne Dilshan (SRI) 319 10,008 22