David Cushley put the Whites ahead on 19 minutes but Don Tierney levelled on 33 minutes before Stephen Hughes put the home side ahead four minutes later.
Gary Thompson scoed a vital equaliser in first-half injury-time.
Darragh Hanaphy missed a Newry chance and Whites keeper Billy Brennan made a great double-save before Scott Davidson's 79th-minute clinching goal.
Newry had been favourites going into Friday's second leg in the promotion/relegation clash after Tuesday's goalless draw at New Grosvenor.
And they dominated the opening half hour of the second half with Hanaphy blasting over from a tight angle after he had rounded the Whites keeper on 65.
Eight minutes later, Brennan made a brilliant double save to deny Matt Haveron and Hanaphy as Newry continued to dominate.
However against the run of play, substitute Davidson struck Distillery's third goal on 79 minutes after a Neal Gawley break.
The goal seemed to totally deflate the home side and the Whites held on to retain their Premiership status.