The hosts seemed to be cruising to their target of 163, adjusted by the Duckworth-Lewis method, before off-spinner Sunil Narine struck three times in an over to bring West Indies right back into the match.
Sri Lanka still needed 11 runs when they lost their ninth wicket and it was left to tail-ender Ajantha Mendis to post his highest ODI score of 21 not out to see his team over the line with seven deliveries to spare.
Earlier, Dilshan (59) pulled and hooked the fast bowlers with disdain, hitting six fours and three sixes while bringing up the fastest ODI fifty of his career in 25 balls.
Sri Lanka were well set at 89 for one until a flurry of wickets and some magic from Narine gave them a fright.
The home team planned to negotiate Narine's overs without taking too much risk but the spinner, back in international cricket after a 14-month absence due to a variety of reasons including work on his action, ran through the middle order.
West Indies were hampered by the absence of Andre Russell, who was injured after delivering five balls, and it meant the touring side had to use part-timers to fill up his quota.
Johnson Charles, bowling the penultimate over, overstepped for a no-ball and Mendis hit the resultant free hit for a six over long-on.
After a lengthy rain delay, West Indies made a poor start to their innings after being put in by Sri Lanka captain Angelo Mathews.
Lakmal bowled an impeccable line and length to reduce the visitors to 24 for three before another interruption meant the match was curtailed to 26 overs a side.
West Indies came out all guns blazing on the resumption with Russell smashing 41 from 24 balls and skipper Jason Holder bludgeoning 36 from 13 to lift his team to a total of 159 for eight.
The second game in the three-match series is at the same venue on Wednesday.
(Writing by Sudipto Ganguly in Mumbai; Editing by Tony Jimenez)
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