Three second half goals gave Linfield a deserved victory at Stangmore Park.
After a goalless first period, Aaron Burns’ turn and shot broke the deadlock, Jimmy Callacher and Andy Waterworth adding further goals.
Inside four minutes Dougie Wilson clattered into Jamie Mulgrew, drawing a yellow card from the referee.
The first shot on target came from Swifts teenager Conor Mullen, but his low, left-foot strike was straight at the goalkeeper.
Despite some impressive possession play, it took 38 minutes for Linfield to test Andy Coleman.
Glenn Belezika’s pass was touched on by Burns to Lowry, but his strike was brilliantly tipped away by the goalkeeper.
Waterworth then saw a shot on the turn go over as the first half ended in stalemate.
Having already lost Kris Lowe in the first half, Dungannon were forced into another change four minutes into the second period when Wilson departed.
And they fell behind in the 54th minute. Grant McCann supplied the corner from the right side and Burns finished off at the back post.
Linfield wrapped up the points in the 70th minute from a Callacher header.
He powered home a header from six yards after getting on the end of a terrific McCann cross.
Waterworth added a third late on when he headed home from Quinn’s cross.