Portadown had Robert Garrett and Chris Casement sent-off in the closing stages as their bid to secure a European place suffered a setback with this draw.
Ex-Ports striker Gary Liggett headed the Swifts in front on the half hour after meeting a Ryan Harpur corner.
Casement levelled in the 63rd minute when his fizzing shot flew past visiting goalkeeper Alvin Rouse.
Garrett was sent-off for a stamp on Jamie Glackin, with Casement dismissed for kicking the ball against Glackin.
Ronnie McFall's team began the match brightly and Peter McMahon saw his shot from outside the box parried by Rouse, who also blocked Casement's attempt with his legs.
At the other end, David Miskelly twice thwarted Terry Fitzpatrick, before Liggett edged Dungannon ahead.
After half-time, Garry Breen headed wide of the left post from a Casement corner as the hosts searched for an equaliser.
Casement did bring his side level with a well-taken goal, then Jamie Douglas fired over the bar as the home side tried to restore their advantage.
Garrett was adjudged to have stamped on Glackin while he was on the ground and Casement appeared to kick the ball against the same player during an ensuing melee.
Both players are now set to miss next week's Irish Cup semi-final against Ballymena United at the Oval.