The Blues outshot the Red Wings 15-1 in the final 7:40 of the second, scoring twice on the power play in 1:39.
"We didn't flip the switch until it was too late," Allen said. "Once we got going, we controlled the game, but you can't take 30 minutes off and expect to win games in this league."
Perron split Glendening and Danny DeKeyser before beating Howard at 13:37 to cut the margin to 3-1, and Vladimir Tarasenko made it 3-2 at 15:16.
"That's not a spot where I would normally attack, but the power play was almost over and I saw a little bit of space," Perron said. "I was just trying to make something happen."
Perron tied the game 3-3 with his second goal of the game at 8:04 of the third period, beating Howard off a pass from Robby Fabbri.
"Once we tied it, I thought we were going to keep rolling and win the game," Perron said. "Obviously, that didn't happen."