Blair Riley hit two of Belfast's goals with Chris Higgins, Mike Forney, Steve Saviano and Alex Foster also on target.
After Higgins' opener, Forney doubled the lead early in period two and while Mike D'Orazio replied, goals from Saviano and Riley left Belfast 4-1 up.
Michal Dobran and Ian Schultz replied for Edinburgh but late goals from Foster and Riley sealed Belfast's win.
After taking one point from Saturday's 3-2 over-time defeat by Fife Flyers, Belfast's weekend return keeps them in third place in the table although they are 12 points behind leaders Cardiff Devils.
The Giants, who do have two games in hand on the Devils, will be back in home action at the SSE Arena against Sheffield Steelers on Saturday night.
Belfast Giants ended a four-game losing run in the Elite League as they clinched a 6-3 away win over Edinburgh.