Craig Peacock put Belfast ahead before Paul Phillips levelled for Manchester prior to the end of the first period.
Mitch Ganzak restored Belfast's lead only for Vinny Scarsella to level.
However, goals from David Rutherford, Colin Shields, Kris Beech and Michael Forney ensured Belfast's win.
The victory sets up a semi-final against either Dundee Stars or Cardiff Devils.
The Giants return to Elite League action at the weekend when they face Fife on Saturday away before a home game against Nottingham Panthers on Sunday.
Belfast Giants advanced to the semi-finals of the Challenge Cup as they beat Manchester Storm 6-2 at the SSE Arena on Tuesday to complete an 11-4 quarter-final aggregate victory.