In a first half of very few chances, it was Linfield who came closest to breaking the deadlock.
Dane McCullough's deep cross from the left was cleared by Kyle McClelland as far as Kirk Millar on the edge of the box, who curled a half-volley that Peter Morrison was equal to as he tipped the ball around the left-hand post.
Minutes later, Kieran Offord almost caught Morrison out who took too long to get rid of the ball, but was unable to hang on to it.
The Blues upped the intensity early in the second half and got their reward 10 minutes after the restart as McGee rose highest to head home Adam Frizzell's corner from the right-hand side.
Cliftonville came close to levelling 10 minutes later through Liam McStravick, whose right-footed strike wasn't far away.
Substitute Joe Gormley went close to finding the bottom corner late on for the Reds, but couldn't keep his strike on target.
Kobei Moore had the chance to put the icing on the cake in injury time but couldn't find a way past Morrison who stood strong.