The off-spinner shrugged off concerns about his bowling action to grab four wickets as Pakistan, electing to field after winning the toss, kept Sri Lanka down to 255-8 in 50 overs.
Hafeez then hammered a stroke-filled 103 to lead the tourists past the target in 45.2 overs, giving them a 1-0 lead in the five-match series.
Ninth-ranked Pakistan need to win the series to stay in contention for a place in the eight-nation Champions Trophy in England in 2017. Sri Lanka, currently ranked fifth, are assured of taking part in the tournament.
Hafeez, who was reported for a suspect action during the preceding Test series and underwent tests in Chennai last week, took to the field because he is allowed to play till the results are declared.
The 34-year-old seized the opportunity to finish with his best one-day figures of 4-41 and to restrict Sri Lanka in good batting conditions.
Hafeez returned with the bat to strike 10 boundaries and four sixes in his 10th one-day century — the last four all coming against Sri Lanka.
The all-rounder, later named man of the match, said he had put aside the scrutiny of his bowling action. “I have definitely put that aside and tried to concentrate on what is in my hands,” he said. “This series is important for us and we wanted to have a good start. I am glad I could contribute to this win and hope I can continue in the same way in the remaining matches. There was no pressure on me. I was given the green signal from the management to play aggressively and the plan worked.”
Meanwhile, Sri Lankan captain Angelo Mathews admitted Pakistan outplayed his side. “We did not play our best cricket,” he said. “We just could not put up a good score or pick up wickets at crucial stages. Hopefully we can come back in the next few games. We need to prepare well mentally.”
Pakistan lost captain Azhar Ali and opening partner Ahmed Shehzad by the 15th over with the score at 65, before Hafeez settled in to fashion a rescue.
He put on 58 for the third wicket with Babar Azam (25) and 75 for the fourth with Shoaib Malik, who ended the match with a six off Tillakaratne Dilshan to bring up his half-century.
Hafeez reached his century in the 36th over with a four off Thisara Perera, but fell two balls later when he gave the same bowler a tame return catch.
Earlier, Dinesh Chandimal top-scored with an unbeaten 65 off 68 balls as the hosts struggled to come to terms with the accurate Pakistani bowling.
Chandimal put on 82 for the fifth wicket with Mathews (38) and 40 for the seventh with debutant Milinda Siriwardana (22) after the first four batsmen had fallen for 118 by the 27th over.
The second one-dayer will be played there on Wednesday.
