Gore, who is in his 12th NFL season and second with the Colts, went into the game with 12,293 yards and needing just 20 yards to move past Brown (12,312).
"I'll be honest, because I'm playing now, I'm really not thinking about it because I am still active and I want to be successful here," Gore said about passing Brown. "I want to do whatever it takes to help this team win."
He got the yardage necessary when he gained 16 yards on a run to the left side with about a minute left in the first quarter. The run gave him 30 yards on three carries in the first quarter.
"Forget pretty impressive -- it's mighty damn impressive," Colts linebacker Robert Mathis said of Gore earlier in the week. "He's in the presence of greatness. He truly does come to work with the intention of getting better working hard, and he's that throwback, hard-nosed, classic type of running back. He earns everything he gets."
Next up on the list for Gore on the all-time rushing list is Tony Dorsett, who has 12,739 yards.
"We gave Frank a game ball," Colts coach Chuck Pagano said. "Frank is a warrior, he is a grinder, a hard worker, a great talent, but he is a great teammate. He loves this game, he loves this team and he puts them on his shoulder week in and week out."
Gore finished Sunday's 29-23 victory with 75 yards on 14 carries to give him 12,368 career yards.