"It wasn't pretty, but we got a point," Benn said. "I guess that's a positive. I like how we found a way to score late there, but this overtime is killing us right now."
Fabbri gave the Blues a 1-0 lead with his first goal in six games with a snap shot to the short side at 13:36 of the first period.
Pietrangelo made it 2-0 with a power-play goal at 4:17 of the second period.
Oleksiak beat Allen with a shot to the short side with 3:39 remaining in the second to make it 2-1, and Benn tied it 2-2 with a power-play goal 1:47 into the third with a quick shot after skating into the right faceoff circle.
Perron gave the Blues a 3-2 lead at 13:12 of the third with his seventh goal. He has three goals and eight assists during an NHL career-high eight-game point streak.
The Stars pulled Niemi, and Oleksiak's shot deflected off Pietrangelo and past Allen to tie it 3-3.
"It's frustrating; it just seems like we can't get any bounces," Pietrangelo said. "We all tried so hard to clear the puck right before that, but nobody cleared it, and those always come back to haunt you.
"I have one finger to hit left, and it hits my pinkie and finds a way to change direction."
The Blues will be looking to find a way to hold on to leads.
"There's real concern about the leads," coach Ken Hitchcock said. "There's real concern with what we gave up and how we did it. We played terrific and had full control of the hockey game, and we got careless on our checking and puck management."
