A late rally by bottom-placed Ards failed to find an equaliser as Darren Boyce's early strike gave Dungannon a third 1-0 home win of the season.
Good work by Fra Brennan created the chance for former Coleraine man Boyce cut cut inside before beating keeper Graeme McKibben with a left-foot shot.
Ards felt they could have had a late penalty but Andy Mitchell was booked for diving instead.
Then Paul McDowell sent a glancing header wide in stoppage-time.
"He probably should have scored that but it went past the post and that is when we knew it was going to be our day," said Swifts manager Darren Murphy.
"It was not a pretty 1-0 win but the result is the most important thing and we were lucky enough to get the win.
"Ards pushed hard for the last 10 minutes and they had a couple of good chances."