Dungannon made it nine draws from 17 Premiership matches with this result.
Josh Cahoon touched the ball past David Miskelly from the edge of the box for his eighth of the season on 16 minutes.
Darren Murray levelled from the penalty spot on 24 after Niall Morgan brought down Ross Redman inside the area.
Morgan saved from Kevin Braniff with his feet while Neil McCafferty also tested the visiting goalkeeper. Cahoon went close with a 30-yard volley and Redman cleared an effort off the line.
The Swifts had Matt Hazley stretchered off in the second period.
Portadown remain in eighth place while Dungannon move up to ninth.