Ulster looked poised to close out a hard-fought win before Finn's long, looping pass found the hands of DTH, who had come off the bench to make his first appearance since injuring his hand last month.
The Canadian winger went over in the corner to level the score and it was left to Finn to nail the tricky conversion to edge Glasgow ahead.
Finn told glasgowwarriors.org: "When we were behind by five points, there was some doubt in the back of my mind but the forwards were amazing to keep the ball and create the space for DTH.
"To come back from injury like he has and to finish that move under so much pressure is just testament to how good a player he is.
"It's so fitting that he scored that try.
"When I was lining up the kick I was just thinking about how I didn't want it to go to extra time as my legs didn't have much more in them.
"I wasn't too nervous when I stepped up and was just so relieved to see it go over."
Glasgow can now watch watch Munster and the Ospreys battle it out for the right to join them at the Kingspan Stadium final next weekend.
Finn added: "I've not thought too much about (the final) as we don't know who we're playing yet.
"We know we need to play to our full potential for the 80 minutes and we're hopeful we can lift the trophy."