They doubled the lead with a stunning goal from McGavery on the counter-attack soon after.
Raiders top-scorer David Dinkin halved the deficit with a last minute consolation.
Maccabi wonder kid Louie Basso was terrific in midfield but he was sent to hospital following a head injury which required stitches.
Manchester joint-manager Stuart Delmonte told JC Sport: "We bossed it for the first 20 minutes without scoring. It was a good performance and I'm just pleased to go through on 3G.
"This match was a great advert for Jewish football. Raiders are a very nice bunch of boys and we now look forward to having Michael Sacks and a couple of the boys back for the quarter-finals."
Raiders boss Rob Richman told JC Sport: "I thought both teams contributed to what was a really decent game of football. Manchester dominated the possession but we probably had the better chances in the 90 minutes.
"It was a very even game and with 15 minutes to go, and the game still goallessm I could see it going to extra time.
"Unfortunately it wasn't to be today, but we gave absolutely everything against a very talented Manchester side.
"Congratulations to Manchester who I believe have a huge chance of retaining their trophy."
