Patrick Kane had a goal and an assist, Alex DeBrincat and Brandon Saad scored and Erik Gustafsson had two assists for Chicago (16-22-7). Collin Delia made 39 saves.
"I think they took it to us in the third period," said Kane, who has seven goals in his past six games and 18 points (nine goals, nine assists) in his past 11. "We were in a good spot there after two, being tied up, and we should feel confident in that position. So it was unfortunate to see them get an early goal, and I think we got away from our game plan a little bit after that."
Gaudreau's 25th goal of the season gave the Flames a 1-0 lead at 3:24 of the first period on a rebound of Monahan's shot near the top right of the crease.
Kane tied it 1-1 at 16:34 of the first on a backhand shot from the goal line that deflected off the top of Rittich's left shoulder.
DeBrincat's one-timer from Kane gave the Blackhawks a 2-1 lead at 5:07 of the second period.
"We need points and we need to get them every night, and that's what we're battling for," said DeBrincat, who has nine points (six goals, three assists) in his past nine games and has reached 20 goals in each of his first two NHL seasons (28 in 2017-2018). "Overall I thought we played alright, but [we] didn't get any points so it doesn't really help us."
Monahan tied it 2-2 with a power-play goal at 18:42 of the second from the low slot off a pass from Matthew Tkachuk. The assist gave Tkachuk his 50th point of the season, an NHL career high.
Gaudreau's second goal of the game gave the Flames a 3-2 lead on a breakaway 54 seconds into the third period.
Garnet Hathaway's empty-net goal made it 4-2 at 19:10 of the third.
Saad scored at 19:31 for the 4-3 final.