Linfield took the lead on 14 minutes when Baird won the ball back in midfield and passed to Mulgrew, whose looping cross played Fitzpatrick through on goal.
The striker then took a clever touch to travel forward before slotting past stopper Gareth Deane.
Shortly after, Kyle McClean's powerful shot looked like it was going to double the away side's lead, but Harry Lynch blocked the effort.
David Healy's side would extend their lead just before the 30-minute-mark though when Fitzpatrick knocked Kirk Millar's corner into the path of Offord, who converted from close range.
Linfield continued to dominate with Liam Hassin making a vital interception to deny Fitzpatrick a second of the evening before the half-time whistle.
Bangor made a bright start to the second half with substitute Lewis Francis heading wide just seconds after the restart.
Mark Haughey's header was then deflected before, at the other end, Fitzpatrick nodded wide.
Bangor kept pushing with Haughey denied once again by Johns and Harry Lynch unable to keep his shot on target.
However, it was the Blues who would net again before the final whistle with McKee coolly firing past the busy Deane in the 90th minute.