Collins turned a steal into the go-ahead basket and Francis added seven points and seven assists after deciding to play through knee tendinitis with the team short-handed.
"Everybody's got to step up and try to fill the void," Richardson said. "We're missing some key guys right now."
Eddy Curry added 22 points for the Knicks, who won their seventh straight at Madison Square Garden despite playing without starting guard Crawford, who had ankle surgery Friday, plus reserves David Lee (ankle) and Nate Robinson (stomach virus).
"It's going to be hard without Jamal," Curry said, "but if everyone comes in and plays like we did tonight we shouldn't have a problem winning games."
Al Harrington scored 26 points and Monta Ellis had 17 for the Warriors, who have dropped six in a row on the road. Golden State is 1-5 since the All-Star break, falling back in the Western Conference playoff race.
The Knicks, meanwhile, pulled within two games of New Jersey for eighth in the East. New York visits Atlanta on Saturday night.
"We're asking our guys to take a big leap to try to get into the playoffs," Knicks coach Isiah Thomas said. "In order to get into the playoffs, we're going to have to win some road games."
Playing without starters Baron Davis (knee surgery) and Stephen Jackson (toe), the Warriors led 92-90 with 5 1/2 minutes to go. But they went cold from there, going without a field goal for nearly 5 minutes while the Knicks seized control with a 12-0 spurt.
"I think we missed a lot of open shots and our confidence got a little bit down," the Warriors' Mickael Pietrus said. "I think we played hard tonight, it's just a matter of playing a better way and winning."
In a game that was close the entire second half, Marbury followed Collins' go-ahead layup with a jumper on the break, giving New York a 95-92 lead with 3:32 remaining. Marbury then hit a 3-pointer for a six-point lead before Richardson knocked down a jumper to make it 100-92 with 2:16 to play.
Marbury added a pair of free throws with 1:10 left for a 102-92 lead. Golden State finally scored again when Harrington made two free throws with 1:02 remaining.
"Tonight our defense in the fourth quarter was awesome," Marbury said. "The way we have been playing in the fourth quarter at home, we have to take that on the road."
The Knicks led 26-18 after one, then extended the lead to 36-24 on Richardson's 3-pointer with 7:10 remaining in the second. Golden State got back in the game behind reserve Sarunas Jasikevicius, who had nine points and five assists in the final 9 minutes.
The Warriors got within one late in the half, but the Knicks led 54-48 at the break after Marbury's layup with 6.5 seconds left.
Ellis had six points during an 8-1 spurt that gave Golden State a 69-68 lead with 4:16 to go in the third, and the game was tied at 75 after three after the Warriors went 0-for-4 from the line in the final 41 seconds.
Golden State was only 6-of-15 at the line. Ellis was 7-of-22 from the field.
"They just wouldn't go down tonight," Ellis said. "But they hit big shots and we didn't, and they got defensive stops. Tonight was just not my night, tonight was theirs."
Davis continues to progress after left knee surgery. Golden State hopes he can play Monday at Detroit, when it ends its five-game road trip. ... Crawford will be on crutches for a week and will begin rehab in 10-14 days. The Knicks expect him to miss the rest of the regular season but hope he can return if they make the playoffs. ... New York had lost four straight and eight of nine against Golden State.