Having conceded a lead of 410, Sri Lanka's defeat was confirmed after Noman spun a web around the batsmen at the Sinhalese Sports Club ground in Colombo on Thursday to claim a career-best 7-70 as Pakistan romped home.
Angelo Mathews remained unbeaten on 63 but Sri Lanka, who managed 166 in the first innings, were all out for 188 in the second.
While Noman was the wrecker-in-chief with his guile and accuracy, quick Naseem Shah claimed the remaining three wickets to secure Pakistan's biggest Test win in Sri Lanka.
"Very proud to win this series and all credit to the guys and the support staff for working so hard in the last three to four months," Pakistan captain Babar Azam said.
"The way our bowlers bowled here, it was good because there wasn't much for them."
Earlier, Pakistan batted for just two overs on day four, allowing Mohammad Rizwan to complete his half-century before declaring on 5-576.
Agha Salman, who was player of the series, remained unbeaten on 132.
Pakistan opener Abdullah Shafique bagged the player-of-the-match award for his career-best 201 that laid the foundation for Pakistan's mammoth total in the only time they batted in the Test.
"We won the toss and chose to bat but 160 is not enough on this wicket," Sri Lanka skipper Dimuth Karunaratne said of their below-par first innings display.
"We knew it's a pretty good and flat wicket but a lot of our batsmen threw their wickets away. That's the turning point.
"After that, we were chasing (Pakistan) all the time."
Pakistan won the opening Test in Galle by four wickets.
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