Roach took 4-51 on Sunday and with fellow seamers Jayden Seales, Shamar Joseph and Alzarri Joseph, combined to end Sri Lanka's innings in only 31 overs, just after lunch on the fourth day.
It is the West Indies sixth-largest win in Tests, and gives them a 1-0 lead in the two-Test series.
Dinesh Chandimal, who made 43, was one of only two Sri Lanka batters to reach double figures. Sonal Dinusha finished on 12 not out.
West Indies set up the win on the third day when Amir Jangoo and captain Roston Chase shared a world record 401-run sixth wicket partnership which allowed the home side to declare at 9-626, with a first innings lead of 318.
The pitch had looked particularly placid on the third day but Roach, Seales and the two Josephs caused the ball to seam and swing on Sunday. Roach, who came into the day with 296 Test wickets, was almost unplayable though the Sri Lanka resistance was thin.
Captain Dhananjaya da Silva, who made 120 in Sri Lanka's first innings of 308, was out for a seven-ball duck.
The 37-year-old Roach bowled Asitha Fernando to claim his 300th wicket and leave Sri Lanka 9-84 as victory rapidly approached.
He is the fifth West Indian after Courtney Walsh, Curtley Ambrose, Malcolm Marshall and Lance Gibbs to reach the 300-wicket milestone.
"It has been a long journey," Roach said. "Well done to the team and Roston Chase who had his first win as captain and first win for Daren Sammy as coach.
"I wouldn't be here if not for these guys. Thanks to my family and I know my mum is watching. She's proud. Thanks to everyone, I really appreciate it and I'm glad I made you proud in some way."
The West Indies' quicks left no doubt about the result when Sri Lanka limped to 6-81 at lunch.
Resuming on 1-15, Sri Lanka lost both overnight batters in the space of four balls when Roach had Nishan Madushka pinned leg before wicket with his fourth ball and Jayden Seales found the leading edge of nightwatchman Kasun Rajitha with his first delivery from the other end.
Shamar Joseph then used the favourable overcast bowling conditions to perfection when his impeccable swing undid Kamindu Mendis, who left a delivery that swung back enough to knock back the off stump.
He then had de Silva trapped lbw for a seven-ball duck.
West Indies had lost seven of their eight Test matches in the World Test Championship but have now moved off the bottom of the points table.
The second Test begins at the same venue on Friday.
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