When the first goal went the way of the visitors, it was against the run of play in the 17th minute.
McClelland's fine volleyed finish from inside the area came after Bangor failed to deal with a Rory Hale cross.
After taking the lead, the Reds looked increasingly threatening on the counter-attack with Bangor having a few half chances of their own up the other end.
Just as it looked like the north Belfast side were going in at the break a goal to the good, they doubled their lead right on the stroke of half-time.
It was Quinn who hooked the ball home from an acute angle after his own effort struck the upright following a deep cross from Oliver Samuels which Quinn met with his head.
Cliftonville captain Hale came close on a couple of occasions to finding a third but was denied by Gareth Deane multiple times.
Substitute McStravick put the icing on the cake for the Reds as he found the top corner from the edge of the box in the 80th minute with the pick of the goals from Clandeboye Park.