Rutherford was unbeaten on 54 from 40 deliveries and Jason Holder clubbed three sixes in the penultimate over, finishing 21 not out off five balls as West Indies reached 5-170 in reply to Sri Lanka's 169 on Sunday.
Two costly missed catches during Rutherford's innings hurt Sri Lanka's chances of taking the series after the first two games were split.
West Indies went into the deciding game with an attacking focus, deciding to bowl first after winning the toss.
Shamar Joseph (5-33) backed up the decision by snaring five wickets to contain Sri Lanka's innings in a player-of-the-match performance.
Joseph took two wickets in the sixth over and three in the last over of Sri Lanka's innings — twice missing hat-trick opportunities.
Matthew Forde took a stunning one-handed catch off his own bowling to remove Kusal Mendis (5) on the ninth delivery of the match.
Sri Lanka accelerated to be 1-49 after five overs, then Joseph struck with wickets on consecutive balls in the sixth.
Pathum Nissanka hit 26 from 17 deliveries until he top-edged a catch to Roston Chase to end a 43-run second-wicket partnership. Joseph then trapped Pavan Rathnayake lbw on the next delivery.
Dunith Wellalage anchored a lower-order rally for Sri Lanka that was undone when Wanindu Hasaranga (21 off 13 deliveries) was run out in the 19th over, ending a 49-run partnership for the seventh wicket.
Wellalage was the first of three wickets to fall in the 20th, posting 43 from 28 deliveries in a run-a-minute innings.
After losing Shai Hope on the third delivery of the run chase, bowled by Wellalage, West Indies raced to 52 before losing three wickets for one run in a 12-ball span that swung momentum back Sri Lanka's way.
Rovman Powell (33) combined with Rutherford to revive the run chase, putting on 81 for the fifth wicket until he was dismissed on the last ball of an eventful 17th over.
The over from pace bowler Dushmantha Chameera contained nine deliveries - including a no-ball and two wides - and netted 24 runs for West Indies and the wicket of Powell. It left West Indies needing 36 runs from three overs.
The equation got down to 29 required from two overs. Holder clouted Chameera for three sixes in four deliveries to end the 19th and leave West Indies needing six runs for victory.
