This was England's second victory in three days after their one-day international success away to Morgan's native Ireland in Dublin on Friday as they continued an encouraging start to a season that sees them staging both the 50-over World Cup and the Ashes.
England fast bowler Jofra Archer also continued his push for a World Cup call-up by taking 2/ 29 in Pakistan's innings.
The recalled Ben Duckett fell early in England's chase, driving Shaheen Afridi to mid-off, but Vince and Root kept England up with the run-rate thanks to a series of sparkling shots.
Vince was caught down the legside by Pakistan captain and wicket-keeper Sarfraz Ahmed off Imad Wasim but Morgan kept the runs coming, taking 14 off three successive balls from medium-pacer Faheem Ashraf.
And even when Sarfraz caught Root, there was little danger to England's hopes of victory with Morgan hitting five fours and three sixes in his 29-ball stay.
Babar Azam (65) and Haris Sohail (50) shared a stand of 103 after Pakistan had been 31 for two following Sarfraz's decision to bat first on a green-tinged pitch.
Haris hit Chris Jordan for six before Babar struck successive sixes off spinner Joe Denly to complete a 31-ball fifty.
Pakistan then saw both their well-set batsmen fall in quick succession with Barbados-born Archer, renowned for his form in T20 franchise cricket, removing Haris before he ran out Babar.
England and Pakistan will now play a five-match ODI series starting at The Oval on Wednesday.