New York scored twice in the 12th after Minnesota had taken the lead in the top half, allowing the Yankees to even the AL Division Series at a game each.
The best-of-five series resumes Friday in Minnesota.
The Twins tied it at 5-5 with a two-run eighth, after which Minnesota's bullpen pitched 4 2/3 innings of hitless relief.
Torri Hunter, who had three hits and scored three times, hit a two-out homer in the top of the 12th off Tanyon Sturtze.
But with one out in the bottom half, closer Tom Nathan walked Miguel Cairo and Derek Jeter. Alex Rodriguez, who went four for six with three RBI, then drove in Cairo with a ground-rule double. Jeter would have scored had the ball not bounced into the stands.
Matsui hit a line drive that Jacque Jones caught in right, but Jeter was able to beat the throw home.