Rajapaksa smashed an unbeaten 71 to help Sri Lanka weather a top order collapse and reach 170-6 at the Dubai International Stadium.
When they returned, Hasaranga claimed three wickets in one over and Pramod Madushan claimed a career-best 4-34 to bowl out Pakistan for 147 in exactly 20 overs.
Put into bat, Sri Lanka were reeling at 58-5 in the ninth over after their top order wilted early.
Naseem Shah began with a wide but soon redeemed himself by uprooting Kusal Mendis's off-stump with the third legitimate delivery of the day.
Haris Rauf removed Pathum Nissanka and Danushka Gunathilaka in successive overs and Iftikhar Ahmed took a sharp return catch to cut short Dhananjaya de Silva's knock of 28.
With Sri Lanka 67-5 at the halfway stage of their innings, Hasaranga (36) combined with Rajapaksa in their most productive partnership of 58 runs from 36 balls.
Rauf (3-29) removed Hasaranga but Rajapaksa remained unbeaten after an incendiary knock that included three sixes and six fours.
It was largely because of his 45-ball blitz that Sri Lanka plundered 103 runs from the last 10 overs.
When they returned, Madushan claimed two wickets in two balls to rock Pakistan.
Babar Azam was caught at short fine leg, while Fakhar Zaman dragged the next ball onto his stumps.
Madushan returned to dismiss Iftikhar (32) after the batsman had combined with Mohammad Rizwan (55) to revive their innings.
Hasaranga effectively derailed Pakistan's chase when the spinner dismissed the dangerous Rizwan, Asif Ali and Khushdil Shah in the same over.
"We dominated the first eight overs but that (Rajapaksa-Hasaranga) partnership proved crucial," Pakistan captain Babar said.
"As a batting unit, we did not do according to our potential, and with the ball, we could not finish well and conceded 15-20 extra runs."
The victory would taste even better for Sri Lanka, who could not host the tournament at home because of a political and economic crisis in the island nation though they retained the hosts' honour.
