Chasing 236 for victory from their allocation of 50 overs, West Indies entered the final over at Queen's Park Oval needing 13 runs to win.
Sri Lanka turned to Vaas, their most experienced bowler, to deliver the last over, and he restricted the West Indies last wicket pair of Chanderpaul and batting bunny Fidel Edwards to three runs from the first four balls of the over.
But Vaas, whose line and length had been unerring throughout the match, delivered a full length delivery and Chanderpaul drove him through mid-off for four.
Sri Lanka would still have favoured their chances with West Indies needing a six off the last ball when Vaas ran up and delivered a full toss which Chanderpaul duly dispatched over the wide long-on boundary to spark massive celebrations in the stands.
Chanderpaul was undefeated on 62 from 63 balls which included five fours and one six.
The victory gives West Indies a 1-0 lead in the three-Test series which continues at the same venue on Saturday before ending next Tuesday with a day/night fixture at the Beausejour Stadium in St. Lucia.
A record sixth-wicket stand of 159 between Chamara Silva and Chamara Kapugedera had helped Sri Lanka reach a respectable 235 for seven off their allotment of 50 overs, after they were sent in to bat on a hard, true pitch under partly cloudy skies.
Kapugedera struck seven fours and three sixes in the top score of 95 from 117 balls and Silva collected seven boundaries in 67 from 96 balls to get Sri Lanka back on course, after Dwayne Bravo triggered a slump that restricted them to 49 for five in the 15th over.
Then openers Chris Gayle and Devon Smith emerged from an uneasy start to give West Indies a solid foundation of 53 for the first wicket.
Kulasekera made the breakthrough, when he gained a dubious lbw verdict to dismiss Smith for 14, but television replays showed the delivery had pitched outside the line of the leg-stump and umpire Clive Duncan may have erred with his judgement.
Sri Lanka were made to toil for almost 13 overs before they scalped another wicket, as Ramnaresh Sarwan joined Gayle and they added 56 for the second wicket.
But Sri Lanka savoured a purple patch, when they captured three wickets - two to Kulasekera - in the space of seven balls to leave West Indies 110 for four in the 28th over.
ODI newcomer Ajantha Mendis had Gayle lbw for 52 with the last ball of the 27th over, Sarwan was caught behind for 35 off the third ball of the next over from Kulasekera, who also had Marlon Samuels adjudged lbw for a duck off the last ball of the same over.
But Sri Lanka could not turn the screws, as Dwayne Bravo joined Chanderpaul and added 59 for the fifth wicket to put West Indies back on track before he was run out in the 39th over for 36 which included an extraordinary six over extra cover off Ishara Amerasinghe which hit a news agency photographer in her head.
Sri Lanka looked to have collared West Indies, when the home team lost five wickets between the 45th and 49th overs before the real party started with Chanderpaul's mighty blow.
ODI newcomer Ajantha Mendis bamboozled the West Indies batsmen with his clever variations to collect three wickets for 39 runs from 10 overs, and Nuwan Kulasekera bagged three for 43 from 10 overs.
Earlier, Kapugedera joined Silva and they established a new Sri Lanka record for the sixth wicket in ODIs.
Silva however, was dismissed in the 47th over, when he was caught at mid-wicket to become the last of Bravo's career-best four wickets for 33 runs from 10 overs which made him the most successful West Indies bowler.
At this stage, the 21-year-old Kapugedera was eyeing triple figures, but he just failed to cross the threshold to his maiden ODI hundred in his 30th match, when he was adjudged lbw swinging across the penultimate ball of the innings, a slower delivery from Fidel Edwards.
The two Sri Lankans had batted with aplomb, after the top-order batting caved-in under the pressure of steady, if not menacing West Indies bowling.
They took few risks, and in the closing overs, their confidence was such that they started to play with greater enterprise before their downfall.
But Sri Lanka had stumbled through the opening phase of the match, when Edwards bowled ODI newcomer Mahela Udawatte for a duck in the second over, and Jerome Taylor had Kumar Sangakkara caught at square cover for 23 to leave them 38 for two in the 10th over.
Sri Lanka appeared to have conceded the advantage, when he snared Upul Tharanga for 10, Mahela Jayawardene for one, and Tillakeratne Dilshan for two between the 11th and 15th overs.
But Kapugedera joined Silva, and they breathed life back into their side's innings with controlled batting once Taylor, Edwards, and Bravo were removed from the attack.
Kapugedera and Silva milked the bowling of Darren Sammy and beanpole left-arm spin bowler Sulieman Benn, playing in his first ODI, as well as West Indies captain Chris Gayle for precious runs.
Kapugedera reached his 50, when he flicked Edwards to deep fine leg for a single in the 40th over, and Silva joined him on the milestone, when he sliced the same bowler to the backward point boundary in the same over before they both perished making the final thrust.