Goals in the second half by Chris Morgan and Stephen Garrett secured a win which boosts Newry's chances of avoiding the relegation play-off.
City had trailed to a sixth-minute goal scored by Paul Brown who shot past home keeper Andy Coleman from 12 yards.
Morgan side-footed the leveller at the far post in the 57th minute after a Darren Cassidy corner had been flicked on by an Institute defender.
Garrett then outpaced defender Paddy McLaughlin to fire in the winner.
It takes one-from-bottom Newry to within one point of Glenavon.
The defeat is a huge blow to Institute's hopes of getting a top six place before the Premiership split on Saturday.
They need to win away to leaders Glentoran and hope that Cliftonville lose at home to Bangor.
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