Alexander Radulov gave the Stars a 1-0 lead with a power-play goal at 4:09 of the first period. Sergachev tied it 1-1 with a power-play goal at 12:58, and Killorn scored on a power play to make it 2-1 at 15:24.
"We got a lot of chances, we've got to just bear down and score goals," Sergachev said. "That's our problem. Besides that, we played good enough."
Palat made it 3-1 with a redirection of a shot by Sergachev at 3:22 of the second period.
Radek Faksa got Dallas within 3-2 at 17:57. Dickinson tied it after a shot by Andrew Cogliano got through Vasilevskiy's pads and he was able to jam it in from close range.
"I saw they were changing," Dickinson said. "I tried to penetrate to the middle and was able to make a move on the first guy. I was just thinking 'Get to the net' in case it might squirt out, and it did."