Dungannon find themselves three points adrift at the bottom of the JJB Sports Premiership after being held at home to Institute on Boxing Day.
Institute took the lead in the 26th minute when right-back Sean Roddy accepted a throw from Kevin Ramsey and curled a fine shot past Dwayne Nelson.
But the Swifts equalised eight minutes later through Mark McAlister.
Marc McCann supplied the pass for McAllister to lift the ball over keeper Michael Doherty and in off the bar.
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