Kubalik pulled the Blackhawks within 3-2 with his second power-play goal at 8:05 on a one-timer from the top of the right circle.
Chicago has scored at least one power-play goal in its first four games for the first time since the 2012-13 season.
"Obviously, I feel that's kind of my wheelhouse, but I'm just happy to score, happy the power play was going pretty well," Kubalik said. "I'm just really happy it worked out this time."
After Bobrovsky stopped Patrick Kane on a penalty shot at 14:28, Kane tied it 3-3 at 19:10 with a high backhand from a sharp angle after getting around Stillman.
The Blackhawks lost by fewer than three goals for the first time in their four games. They were swept by the Tampa Bay Lightning 5-1 and 5-2 to start the season.
"Well, it's hard to talk about it, obviously," Kubalik said. "We wanted to win, but at least we get a point. I think the work ethic was pretty good today and felt like we did lots of good things. We finally get some goals too. Just sad that we didn't make it to the win."