England won the toss and elected to field in a match reduced to 21 over per side because of rain at Pallekele. Sri Lanka made 150-9 in its 21 overs with Niroshan Dickwella top scoring with 36.
Rashid returned 4-36 in five overs while Tom Curran had 3-17 off four overs.
England reached their target for three wickets and 15 deliveries remaining. Morgan made 58 not out off 49 deliveries including seven boundaries while Jason Roy made 41.
Sri Lanka got off to a brisk start as Dickwella and Sadeera Samarawickrama shared 57 runs for the first wicket by the sixth over. Dickwella was first to be dismissed for 36 in 20 deliveries caught by Chris Woakes off Curran. Dickwella hit eight boundaries.
Kusal Mendis was out first ball, bowled by Rashid. Samarawickrama made 35 before being caught by Woakes off Rashid.
English bowlers picked up wickets regularly thereafter to restrict the home team to a below par score.
England lost two quick wickets in their chase with opener Jonny Bairstow (4) and Joe Root (8) departing early.
Left-arm spinner Amila Aponso picked up both wickets having Bairstow caught by Thisara Perera and bowling Root with straight delivery to leave England 34 for two.
Roy and Morgan shared 46 runs for the third wicket.
Roy took only 26 deliveries for his 41 runs which included two sixes and three boundaries. He was declared out lbw on review off spin bowler Dhananjaya de Silva.
Ben Stokes remained not out on 35 and shared an unbroken 73-run stand with Morgan.
Morgan credited his bowlers for the victory.
"Sri Lanka came out fighting with the bat earlier on, knocked us back a little bit but three of our bowlers were outstanding today, he said. "If (the batters) get an opportunity to put away a bad ball and impose themselves on the game, they have full license to do that, that's the way they play."
Sri Lanka's Perera said his team's plan of trying to put England under pressure by getting a big score may have misfired.
"We were thinking we have only 21 overs so we have to get more runs on the board otherwise we can't put pressure on the England team. Actually we were 30, 35 runs short."
The first game ended in no result due to rain and the fourth match will be played at the same venue on Saturday.