Opting to bat first, Sri Lanka could manage only 178 for nine in the allotted 50 overs at the JSCA International Stadium Complex.
18-year-old Deepti Sharma was the wrecker in chief, picking up four for 23 with her medium pace bowling, to set up the home side's win.
Skipper Mithali Raj then remained unbeaten on 53 to guide her team home with as many as 41 balls to spare. India were 179 for four in 43.1 overs when the winning run was scored.
Mandhana hit four boundaries and a six in her run-a-ball innings, while Kaur found the fence six times while facing 61 balls.
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Chasing a modest total, India were off to a solid start scoring 67 runs for the opening wicket in just under 14 overs before Thirush Kamini fell to Sugandika Kumari for a 44-ball 26, which included five hits to the fence.
Going through a good run of form, Mandhana too got out to the same bowler when in sight of yet another half century.
Left-arm spinner Kumari was the most successful bowler for Sri Lanka, returning figures of 4/39.
Earlier, the Lankans lost wickets at regular intervals to be restricted to a meager total.
Dilani Manodara top-scored for the visitors with an unbeaten 51-ball 43, hitting five boundaries in the process, while opener Prasadani Weerakkody contributed a painstaking 37 facing 80 deliveries. The rest could not pose much of a fight.
Sri Lanka women: 178/9 in 50 overs (Manodara 43, Weerakkody 37; Sharma 4/23)
India women: 179/4 in 43.