Swifts substitute Joshua Cahoon netted in the second minute of injury time to deny Portadown a Premiership victory.
Stephen O'Neill and Adam McMinn missed the target for Dungannon in a poor first half while Ross Redman's effort was well saved by Johnny Curran.
The visitors dominated after the break and a rasping Darren Murray strike gave them an 80th minute lead.
Dungannon grabbed a point when Terry Fitzpatrick headed on for Cahoon to fire home a last-gasp equaliser.
The point takes Dungannon up to third in the standings with Portadown in sixth.