Robinson Cano tripled off Brian Duensing in the second inning and scored on Jorge Posada's single, Mark Teixeira added an RBI single in the third and Thames and Nick Swisher homered later.
The wild-card Yankees begin the ALCS on Friday at Texas or Tampa Bay. With the short first-round series, the defending World Series champions put ace CC Sabathia in position to start the opener.
The AL Central champion Twins lost their 12th straight postseason game dating to 2004, one shy of the record set by the Boston Red Sox from 1986-95.
In addition to this year, the Yankees knocked out the Twins in the first round in 2003, 2004 and last season. New York outscored the Twins 17-7 in this series and scorched them 69-36 in the four playoff triumphs,
With Minnesota badly missing Justin Morneau, sidelined since midseason because of a concussion, the heart of the Twins' order Saturday, AL MVP Joe Mauer and Jason Kubel, combined to go 3 for 20 (.150) with no RBIs in the series.
Rays 6, Rangers 3
ARLINGTON, Texas (AP) — John Jaso hit a tiebreaking RBI single in the eighth after Carlos Pena had another rare clutch playoff hit for Tampa Bay, and the Rays avoided elimination in the AL division series by beating the Texas Rangers.
The Rays, the AL's best team in the regular season, cut their deficit in the best-of-five series to 2-1. Game 4 is Sunday.
Limited to a total of one run while losing the first two games at home, Tampa Bay broke loose in the late innings.
With the record crowd of 51,746 still buzzing from Ian Kinsler's leadoff homer in the seventh that put the Rangers up 2-1, Dan Johnson doubled off the wall with one out in the Rays eighth. Pena followed with an RBI single that tied the game.
After B.J. Upton struck out, reliever Darren O'Day was pulled after facing only one batter. Rangers manager Ron Washington made a curious move bringing All-Star closer Neftali Feliz into a tie game in the eighth. Feliz walked Jason Bartlett, the No. 9 hitter in Tampa Bay's order, before Jaso's liner gave the Rays their first lead in the series.