Stephen O'Flynn scored his third league goal of the season to give Institute victory as the Drumahoe outfit moved up to fourth position in the Premiership.
Striker O'Flynn steered in a header from Declan McKeever's pinpoint cross three minutes before half-time and Darren McAuley could have made it two.
Institute goalkeeper Ruairi Kelly made fine saves from Padraig Smith and Jonathan Breen on his senior debut.
Warrenpoint remain without a point from their opening five league matches.
In the first half, Smith's backpost header was well saved by Kelly, while Stephen McCabe drilled a long-range shot narrowly wide for the visitors.
Graham Crown made a vital block to deny Stephen Hughes, before O'Flynn grabbed the crucial goal.
Breen passed up a gilt-edged opportunity to equalise but his side-footed effort from close range was saved by Kelly.