The Sri Lankan pair took few risks after the West Indies declared at tea on 477 for nine in reply to the tourists' first innings of 354.
Sri Lanka made 126 without loss in their second innings as play was concluded.
West Indies captain Brian Lara had lit up the final day of the rain-hit match reaching a majestic 209, but with a day and half of play lost to rain, it came too late to put Sri Lanka under any pressure.
Lara admitted that the circumstances of the match had left him to rate the innings behind his four other double centuries.
"It falls behind the others but the important thing is that it is in the present and it builds up for a good ending to the series in the second Test in Jamaica," he said
Atapattu, who made 118 in the first innings, marked the end of play by reaching exactly 50, while Jayasuriya was 72 not out.
Sri Lanka captain Hashan Tillakaratne acknowledged that his side would have to improve to stop Lara in Friday's second test, but also identified the middle order as the area that needed to progress.
"After winning the toss we wanted to make a formidable score and 354 on a good batting track was just not good enough.
"We made a few mistakes and we want to rectify this in the second Test," he said.
The West Indies bowlers struggled to draw any life out of a lacklustre Beausejour Stadium wicket in the final session, and Corey Collymore could produce none of the swing that gave him a five-wicket haul in the first innings.
Lara's fifth time past 200 in Tests was laced with 24 boundaries and one triumphant six launched in celebration of his double century.
He had survived a scare early in the day when Muralitharan lured him out of his crease, but Romesh Kaluwitharana missed the stumping chance.
Prabath Nissanka finally ended Lara's innings when he edged the ball to Kaluwitharana for an easy catch.
The West Indies also took heart from a mature 50 by 20-year-old off spinner Omari Banks to keep the runs flowing after the loss of Lara, while Muttiah Muralitharan was the best Sri Lankan bowler with five for 138.
Tillakaratne praised Muralitharan but said Sri Lanka needed to find more depth in the bowling line-up.
"(Chaminda) Vaas and Murali bowled their hearts out ... but our back-up bowlers didn't do all that well," he said.
"We have a few youngsters and I am sure we would want to give them an opportunity in the second Test."
His thoughts suggest that pace bowler Thilan Thushara and off-spinner Kumar Dharamasena may get a chance in Jamaica.