Sharma became only the second Indian to pass 2,000 runs in this format as India chased down their target of 199 with eight balls and seven wickets to spare.
Jason Roy blasted 67 runs off 31 deliveries, including seven sixes, and opening partner Jos Buttler hit 34 off 21.
England had got off to a flier, reaching 43 off four overs and 118/2 off 10 overs before being reeled in by the Indian bowlers and posting a total that was 20 or 30 runs short.
This was India's sixth successive T20I series win.
Rohit stroked his way to 100 not out, his third T20I hundred, with the help of 11 fours and five sixes, adding 89 runs for the third wicket with Kohli, who got his first score of note on the tour of UK.
For England, Chris Jordan and Adil Rashid bowled two good overs, conceding just three and four runs in the 15th and 16th overs, but with three fours off the 17th over, Hardik Pandya and Rohit released the pressure. However, they lost the second game after attacking English batsman Alex Hales struck 12 runs off the final over to register a five-wicket win over the visitors levelling the series 1-1.
After England squared the series in Cardiff last Friday India needed to win to make it seven successive worldwide T20 series win in succession.
A six brought up Roy's half-century from 23 balls - the joint second quickest for England in T20s - and the hosts looked set for a total in excess of 200. The match will not be played under lights either, which may tilt the balance more towards the batsmen. "That was the reason behind it (bowling short)", said Morgan.
"We have a fantastic supporting staff and leaders who give us huge confidence to go and express ourselves. This series for us has produced three different performances". Jason and Jos were brilliant up front, and nearly gave us a licence to allow ourselves to think about 220.
Captain Kohli said: "The comeback from the bowlers was outstanding - I thought they would end up getting 225 or 230 which would have been tough".
However in order to capture the series, Indian openers Rohit Sharma and Shikhar Dhawan will have to bat cautiously and keep their wickets intact. But the character our bowlers showed, that's something we're very proud of. After a good day with the ball, Pandya was handed a promotion to No. 4 with India still 48 away, and he did not disappoint.