The Blues spurned a golden opportunity to extend their lead when Offord was brought down in the box on 21 minutes. Shields stepped up but saw his penalty saved by Carlos Olses, and Ethan McGee fired the rebound high over the bar with the goal at his mercy.
Offord popped up with a moment of magic to double the lead on 34 minutes, as he collected Shields' pass in midfield and fired a superb 25-yard strike into the bottom corner, which gave Olses no chance.
The second half proved a quieter affair in south Belfast and chances were at a premium.
Fitzpatrick saw a shot blocked by Mohamed Youla soon after the restart, while the visitors managed just one effort of note when Liviu Antal dragged a long-range attempt wide.
Chris McKee came off the bench and almost made an instant impact on 73 minutes, but Olses gathered his header from Kyle McClean's cross.
The away side's stopper continued to frustrate the home side late on as he pulled off a fine save to deny Fitzpatrick's header from another dangerous Millar delivery.