Cormac Burke earned relegated Institute a draw against the Swifts with a second-half strike at Stangmore Park.
Dungannon took a deserved lead on 22 minutes when keeper Corey Wilson could only palm Jamie Glackin's corner into his own net.
Gary Liggett headed wide before the break while the visitors improved in the second half and Burke's lob brought them level after 63 minutes.
Swifts substitute Jamie Douglas twice went close to a winner late on.
Institute's Paddy McLaughlin dragged wide with the first decent chance but the Swifts soon gained control and Abiola Sanusi sent an effort wide before Glackin's opener.
Burke's chip eluded Andy Coleman for the equaliser before the Swifts keeper denied Michael McCrudden.
Institute are already planning for life in the Championship while Dungannon remain 10th.