After the start of the day-night match was delayed by an hour and 20 minutes due to persistent rain, further bad weather during the West Indies innings reduced the game to 26 overs-a-side.
Fast bowler Suranga Lakmal claimed three quick wickets as the tourists slipped to 24-3 before the rain break, and then became 42-4 soon after resumption.
Russell smashed three sixes and as many fours in 41 off 24 balls and left-handed Bravo chipped in with 38 during an entertaining fifth-wicket stand to retrieve the situation.
Skipper Jason Holder then hit 36 off 13 balls with three sixes and two boundaries as the West Indies plundered 59 runs in the final five overs.
Sri Lanka swept the preceding Test series 2-0. The three one-dayers will be followed by two Twenty20 games.