Kane passed from behind the Winnipeg net to Ryan Carpenter in the bottom of the right face-off circle, and he sent the puck across the goalmouth to Brandon Saad for a tap-in that made it 4-1 Chicago at 14:14 of the third period.
"Great play all around, Erik Gustafsson] to keep it in, Carpy making the heads-up pass to Saader backdoor to finish it," Kane said. "Nice to get it out of the way, nice to move on. As far as everyone coming onto the ice and sharing that with me is something I'll never forget."
[WATCH: [All Jets vs. Blackhawks highlights]
Kane (31 years, 61 days) is the youngest United States-born player to reach the milestone, ahead of Jeremy Roenick (32 years, 13 days). The right wing has 380 goals and 620 assists in 953 games in 13 seasons, all with the Blackhawks.
Alex Nylander and Gustafsson each had a goal and an assist for Chicago (24-20-6), which won its season-high fifth straight game and is 11-4-0 in its past 15. Robin Lehner made 36 saves for his third straight win.
"Big win for us, obviously, and for Kaner to get his 1,000th point is a huge thing for our team and obviously for him," Blackhawks coach Jeremy Colliton said. "But I think everyone's pretty honored to have been a part of it. It was fun to watch and to get it on a night when we get two big points against a divisional opponent and in regulation, it feels good for everyone."