England, who won the five-match one-day international series 3-1 against the hosts, posted 187-8 in their rain-interrupted innings after being put in to bat first.
Sri Lanka lost the top half of their batting to the England spinners with just 77 on the board and were finally all out for 157 on the last ball of their innings.
Denly, playing his first match for England since a T20 against Pakistan in Dubai in February, 2010, opened the bowling and made an immediate impact by dismissing Kusal Mendis in his first over.
He then picked up the other opener Niroshan Dickwella in his next over.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
The 32-year-old returned in the last over of the match to add two more wickets to his tally to finish with match figures of 4-19 to script a strong comeback to international cricket.
Sri Lanka's batsmen also failed to read much of Rashid's bowling, with his four overs of leg-spin costing just 11 runs for three wickets.
Sri Lanka captain Thisara Perera top-scored for his side with 57 in 31 balls but his knock was too little too late.
Earlier, the 28-year-old Roy hit four boundaries and six sixes in his knock but also had to thank the Sri Lankan fielders who put down three catches he offered.
All-rounder Moeen Ali also made a handy contribution with the bat, plundering 27 off 11 balls.
Sri Lanka's ambidextrous all-rounder Kamindu Mendis showed signs of promise on his debut with the bat in a fearless knock of 24 off 14 deliveries but the 20-year-old went wicketless with the ball and proved expensive, giving away 27 in three overs.
The two sides will now contest a three-test series, starting in Galle from Nov. 6.
(Reporting by Sudipto Ganguly in Mumbai; editing by Ken Ferris)
