After a quiet first 20 minutes at Stangmore Park, James Knowles came close to breaking the tie when his free-kick went just over the crossbar.
Linfield goalkeeper David Walsh was then called into action to make a low save to block Leo Alves.
In the 34th minute, the visitors called for a penalty after Gael Bigirimana brought down Matthew Fitzpatrick just inside the box, but referee Declan Hassan waved away their requests.
Soon after, Dunne did well to deny Cooper’s long-range shot, with Jamie Mulgrew unable to get the rebound on target.
The deadlock was broken in the 53rd minute when Cooper surged forward and sent a low shot into the bottom left corner from the edge of the box.
Linfield came close to scoring a quick second when Mulgrew’s cross was not cleared by the Dungannon defense, giving Kirk Millar time to control the ball and drive powerfully towards goal, with Dunne making an impressive save.
On the other hand, Knowles sent a timely cross into the penalty area for McGovern, whose header from close range went straight to the stopping Linfield Walsh.
Knowles continued to play a key role in Dungannon’s attacks and the midfielder’s free-kick hit the left post as McAree’s men continued to look for an equaliser.
Then, in the 87th minute, home stopper Dunne was shown a red card for holding the ball outside the penalty area as he tried to thwart Fitzpatrick’s chance when he had already scored.
