With the second T20 being washed out, India have now taken a 3-0 lead in the five-match series. The final match will be played on Tuesday. The visitors had also won the preceding ODI against Sri Lanka.
Patil (3/36) helped India reduce Sri Lanka to 134 for five in a shortened 17-over a side contest. Shashikala Siriwardene top-scored for Sri Lanka with 40 off 32 balls.
India chased down the target with more than an over to spare and seven wickets in hand, thanks to Patil (54 not out off 42) and teenager Rodrigues (52 not out off 37).
It was a second successive half-century for the 18-year-old, who had hammered 57 off 40 balls in the third T20.
India were 56 for three after seven overs, needing 79 off the last 10 overs. Patil and Rodrigues completed the task with ease, sharing a 96-run stand off 70 balls. Rodrigues hit five fours and two sixes while Patil's knock comprised seven boundaries.
India had lost three wickets for 41 runs in the first four overs before Patil and Rodrigues came up with a match-winning effort.
Brief Scores: Sri Lanka 134/5 in 17 overs (Shashikala Siriwardene 40, Anuja Patil 3/36). India 137/3 in 15.2 overs (Jemimah Rodrigues 52 no off 37 balls, Anuja Patil 54 no off 42 balls). India lead five-match series 3-0.