The 23-year-old took a career best 5-34 and was ably assisted by leg-spinner Shadab Khan (2-37) as Sri Lanka, who won the toss and batted, lost their last nine wickets for only 106 runs.
Sri Lanka skipper Upul Tharanga (61) and Niroshan Dickwella (18) put on 59 for the first wicket, but from 102-1 in the 26th over their innings was derailed by Pakistan's bowlers.
Shadab, man-of-the-match for his 52 not out and 3-47 in the second game on Monday, had Dinesh Chandimal (19) leg- before and Tharanga caught in the deep.
Hasan, who had given Pakistan the early breakthrough with Dickwella's wicket, removed Chamara Kapugedera (18), Milinda Siriwardana (two) and Jeffrey Vandersay (nought) to complete 50 wickets in his 24th ODI.
The previous record was held by legendary paceman Waqar Younis, who took 27 ODIs to reach the mark.
Hasan's previous best bowling figures were 5-38 against the West Indies in Guyana earlier this year.
Lahiru Thirimanne (28) and Thisara Perera (38) took Sri Lanka past the 200 mark.
Pakistan lead the five-match series 2-0 after winning the first match by 83 runs in Dubai and a 32-run victory in Abu Dhabi.
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