Asif, 23, was instrumental in turning the game around after Pakistan fell 109 runs behind on first innings.
Asif claimed five for 27 as Sri Lanka were bowled out for just 73, and finished with 11 for 71 in the match. It was Sri Lanka's second lowest total against Pakistan.
''A wonderful performance from Asif got us back in the game,'' Pakistan coach Bob Woolmer told reporters.
''When we conceded a 109-run deficit we were looking down the barrel but then Asif started to take wickets and it all started to happen for us.
''The only way Pakistan could win was by bowling out Sri Lanka cheaply but I was looking at a chase of around 250 and to bowl them out for 73 was a shock to us.
''We needed to get through the new ball and then (Muttiah) Murali's first spell - when we did that I was confident that we'd win.'' Pakistan cruised to 183 for two in 43.2 overs to win the match and the series after the first Test was drawn.
Woolmer believes Asif, named man of the series, is a major find. ''It is very exciting for Pakistan that he has bowled as well as he has done here,'' said Woolmer.
''Asif is one of those McGrath-like bowlers who bowl really tight line and lengths and just get the ball to go in and out a little,'' Woolmer said.
''He moves the ball, is difficult for batsman to read and he is not slow. He is the sort of bowler that Pakistan has not had for some time and we know need to look after him.'' Sri Lanka captain Mahela Jayawardene agreed Asif had stolen the momentum from the hosts.
''The last hour last night cost us the match and the series,'' said Jayawardene. ''It's all about momentum and that shifted to them.
''You can't take anything away from him - Asif bowled really well here and at the SSC (in Colombo).
''There was something still there for the quick bowlers this morning but we could not get those early breakthroughs that we needed defending just 182.
''We were confident at the start of play but they started batting well and then got away from us.''