Azhar Ali played a captain's innings of 79 and senior pro Shoaib Malik made a run-a-ball 51, the pair adding 83 for the fourth wicket after the tourists won the toss and elected to bat in the day-night match.
But Rizwan stole the show with five boundaries and a six as the tourists plundered 88 runs in the last 10 overs.
Anwar Ali, who put on 56 for the seventh wicket with Rizwan, returned unbeaten on 29 off 20 balls.
Ninth-ranked Pakistan need a series win to stay in contention for a place in the eight-nation Champions Trophy in England in 2017.
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Sri Lanka, currently ranked fifth, are assured of taking part in the tournament.
Ahmed Shehzad put on 59 for the first wicket with Azhar when he was bowled by debutant left-arm spinner Sachith Pathirana for 30 in the 14th over.
Mohammad Hafeez, the hero of Pakistan's win in Dambulla with four wickets and a century, managed only nine this time when he became Pathirana's second victim to make it 75-2.
Pathirana and sling-arm fast bowler Lasith Malinga claimed two wickets each.