Collier's shot capped a stirring comeback by the Lynx, who trailed by as many as 18 points in the first half and by 15 in the fourth quarter.
With Collier and Courtney Williams leading the way, the Lynx took an 84-83 with 5.5 seconds left in regulation, when Williams was fouled on a three-pointer and made the free throw for the one-point edge.
But on a multi-chance final possession for New York Breanna Stewart was fouled with less than a second on the clock. She made just one of two free throws and they went to overtime.
Minnesota scored the first four points of the extra session and didn't trail, although a steal and layup by New York's Jonquel Jones saw the Liberty pull level at 93-93 with 28.5 seconds left.
Collier answered with a turnaround basket in the lane and Stewart couldn't get her final attempt to fall at the buzzer.
Collier said the key to the comeback was "just taking it one possession at a time".
"We've been in situations where we've been down before, and that's when we really lean on our defense," she said.
"That's something we were talking about, getting three stops in a row, chipping at it a little bit at a time. Not thinking about the point difference, but thinking about the possession, that we need to get a stop and a score."
Williams scored 23 points for Minnesota, whose 18-point comeback matched the largest in WNBA Finals history -- previously achieved by the Liberty in game two against the Houston Comets in 1999.
Kayla McBride added 22 and Collier had 21 for Minnesota while Jones led the Liberty with 24 points and 10 rebounds, Sabrina Ionescu scored 19 and Stewart chipped in 18.
The Liberty, chasing their first title, host game two on Sunday.
The Lynx, who are seeking a record fifth championship, are in the Finals for the first time since 2017.
