Substitute Conor Forker scored an equaliser for Dungannon in added time after Gary McCutcheon looked to have earned Ballymena victory with his 88th-minute strike at Stangmore Park.
Johnny Topley had several first-half chances and Michael Funston and Michael Ward also went close for the Swifts.
Conor Downey, McCutcheon and Alan Teggart all had scoring opportunities.
McCutcheon fired in the opener late on and Forker took advantage of poor defending to equalise from a corner.
Topley sent his swerving effort past the post in the early stages and Funston also shot wide.
Topley missed from six yards and Wayne Drummond then stretched to tip away his 20-yard effort.
Ballymena's best chance fell to Downey but he was off-target with his attempt at goal.
The visitors came into the game more in the second half, with Teggart almost levelling.
The late drama ensured that both sides left with a point.