Martin Murray struck a late equaliser to deny Warrenpoint a shock win over the Premiership champions at Solitude.
Stephen McCabe wasted a glorious chance for the visitors when he fired a penalty over just before the break.
Daniel Hughes pounced on a defensive mistake before rounding Ryan Brown and slotting in to put Town ahead.
Murray headed home from six yards with five minutes left and Reds winger Martin Donnelly saw his last-gasp header cleared off the line.
The were few chances in a low-key first half, with Reds forward Joe Gormley shooting just wide before a Mark Hughes effort failed to trouble Brown.
The penalty was awarded a minute from half-time for a Chris Curran handball from Darren King's cross, and McCabe failed to find the target from 12 yards.
Ryan Catney was a fault for the 77th minute opener, his misplaced backpass allowing Daniel Hughes to nip in and supply a cool finish.
Substitute Murray levelled from close range and the Reds almost won it with Donnelly's header.
A disappointing result for Cliftonville but they remain four points behind leaders Linfield while Town are now only ahead of bottom team Ards on goal difference.