Escobar, who entered as a pinch-hitter in the seventh, connected on a 1-0 pitch from Axford (0-3), driving it over the wall in right for his first homer since June 20 and fourth of his career in 344 at-bats.
It was the second straight loss for Axford, who gave up a three-run homer in the ninth to Chicago's Dayan Viciedo on Sunday as the White Sox rallied for a 4-3 win.
Caleb Thielbar (2-0) pitched a perfect ninth and Glen Perkins struck out two in the 10th for his eighth save.
The Indians, who only got two runners to second base, had been the only team in the majors not to play in an extra-inning game. Cleveland has dropped eight of 10.
Both teams got strong efforts from their starting pitchers.
Minnesota's Kyle Gibson allowed only two hits in seven scoreless innings, and Cleveland's Zach McAllister gave up five hits while blanking the Twins over 6 2-3 shutout innings.
The Twins played without first baseman Joe Mauer, who is day-to-day with lower back spasms.
Gibson has been blessed with an abundance of runs all season. The Twins came in averaging 7.14 runs in his first five starts, but they couldn't push one across against McAllister, who was pitching on regular rest after going on three days' last week in Los Angeles.
McAllister worked out of a jam in the sixth, stranding a runner at third to keep it tied 0-0.
Minnesota's Sam Fuld doubled down the left-field line with one out and then easily stole third when he got a great jump off second against an unsuspecting McAllister. The right-hander, though, buckled down and struck out both Trevor Plouffe and Chris Colabello looking to end the threat.
McAllister punctuated the whiffs by pumping his fist as he walked to the dugout.
He was lifted with two outs in the seventh and tied his career-high with eight strikeouts, a day after Corey Kluber's 13-strikeout performance against the White Sox.
Gibson and second baseman Brian Dozier combined on a sparkling defensive play in the fourth.
Michael Brantley hit a grounder toward right that Dozier was able to stop with a dive before spinning in one motion and throwing toward first. With first baseman Chris Colabello out of position, Gibson alertly covered and snagged Dozier's throw on one hop before sliding into the bag ahead of Brantley.
Gibson walked Carlos Santana, who stole second. But Gibson kept him there by retiring Lonnie Chisenhall and Asdrubal Cabrera on flies to left.
NOTES: Indians OF Michael Bourn missed his second straight game with a tight right hamstring, the same one he had surgically repaired in October. Bourn could be headed to the disabled list for the second time this season if it doesn't respond. ... Cleveland placed DH Jason Giambi on the DL with a strained calf muscle. The 43-year-old is 0-for-10 this season. ... Twins rookie SS Danny Santana made his major league debut as a pinch-runner in the seventh and nearly got picked off second. Indians manager Terry Francona challenged the safe call but it was upheld. Santana got his first big-league hit in the ninth. ... Indians RHP Josh Tomlin will make his first start Tuesday since July 27, 2012, three weeks before undergoing Tommy John surgery. ... Twins RHP Samuel Deduno will start Tuesday after seven relief appearances. He had shoulder surgery in September.