The left-arm spinner took the first seven wickets and fast bowler Naseem Shah took the final three to bowl Sri Lanka out for 188 in the evening session in Colombo.
Noman returned career-best figures of 7-70 in his 15th Test to hand Pakistan their biggest Test win over Sri Lanka and fourth overall in innings victories. Sri Lanka suffered their heaviest Test defeat at home.
"Everyone stepped up and showed hunger to win this series," skipper Babar Azam said.
Pakistan, who won the opening Test by four wickets, declared their first innings on 576-5 after just two overs in the morning session with a blowout lead of 410 runs.
Opener Abdullah Shafique stood out with his 201 in an innings that saw three century stands, including a 124-run partnership with Agha Salman, who smashed an unbeaten 132.
Pakistan adopt 'Bazball'
Runs came quickly for Pakistan's batsmen, whose approach had a touch of "Bazball" — the aggressive style devised by England's coach Brendon McCullum and skipper Ben Stokes.
"Being aggressive was part of the plan, it's something we had discussed before coming here," said Azam.
"We wanted to play positive cricket. So all the credit to the boys. It's not easy to play spin here but they played very well."
Shafique was named man of the match for his maiden Test double ton.
Salman was man of the series for his 221 runs, including a 177-run stand with double-centurion Saud Shakeel in the first match in Galle.
Sri Lanka started strongly with an opening stand of 69 until Noman bowled Nishan Madushka for 33 with his first ball of the day and ruled the roost thereafter.
Skipper Dimuth Karunaratne fell for 41 after lunch and Noman soon sent back Kusal Mendis and Dinesh Chandimal before Dhananjaya de Silva attempted to resist in his brief stay with Mathews.
De Silva fell for 10 when he was caught out by Abrar Ahmed while attempting a big heave over long-on.
Angelo Mathews hit an unbeaten 63 and put up a fight with Ramesh Mendis before he was finally stumped off Noman, who went wicketless in Sri Lanka's first innings total of 166.
Naseem bowled at express pace and reverse swung the old ball to miss out on a couple of close lbw calls but finally wrapped up the tail with two wickets in one over.
Pakistan's concussion substitute Mohammad Rizwan earlier reached 50 off 67 balls with three fours in the second over and put on 108 runs with overnight partner Salman.
Pakistan held all the aces after bundling Sri Lanka out for 166.
"The batting unit needed to stand up but 166 in the first innings wasn't enough," said Karunaratne.
"It was a pretty good wicket but we threw away our wickets and we were always chasing from there on."