Ben Francisco had a home run and Sizemore a gift RBI double for Cleveland, which has lost eight of 12.
"We didn't have much to offer for [Liriano]," said Jamey Carroll, who went 0-for-4.
Matt Ginter (1-3) allowed five earned runs in six innings to lose his third straight game.
"[Ginter] left some pitches out over the plate for them to hit," said Cleveland manager Eric Wedge.
Liriano (1-3), who was 10-0 with a 2.67 ERA in his last 11 Triple-A starts, was called up Friday to replace Livan Hernandez in the Minnesota rotation. He missed last season following Tommy John elbow ligament replacement surgery on his left elbow and was 0-3 with an 11.32 ERA for Minnesota in three starts this April before being demoted.
"He did a good job with his fastball," Wedge said. "He did a good job mixing in his slider and changeup. He kept the ball down, for the most part."
Ginter let his pitches drift up in the middle innings and it again cost him.
"When I've been getting the ball up, that's when I'm getting hurt. If it gets in the middle of the zone, any of the hitters 1-9 are gonna smack it," said Ginter, who allowed a pair of fifth-inning runs, including an RBI triple by Mike Lamb. Nick Punto hit a run-scoring triple in the sixth and scored on a wild pitch to Jason Kubel.
"I wanted to make sure I didn't hang it. I just came back, was choking it a little too much and threw it in the dirt," Ginter said.
In his last start July 29 against Detroit, Ginter left a couple pitches up in the fourth inning that turned into two-out run-scoring doubles, ending his night.
But Ginter isn't solely to blame for the Indians' 14th loss in their past 18 road games.
Again, Cleveland batters had chances to hurt an opposing pitcher but were unable to get the key hit. The Tribe was 0-for-6 with runners in scoring position Sunday, and 6-for-25 in the series.
"We had a lot of chances to get runs," Franklin Gutierrez said. "We didn't do it and they did. We've just got to make sure we're making good swings."
A pair of walks and a throwing error allowed the Indians to load the bases with two outs in the third, but Francisco struck out on a 93-mph fastball from Liriano.
Down, 1-0, after Brendan Harris homered off Ginter in the Minnesota third, the Tribe had runners on first and second and one out in the fourth. Gutierrez, up 3-0 in the count at one time, hit into a 6-4-3 double play to end the threat.
Andy Gonzalez, who got his first start at first base, singled to start the fifth. After Gonzalez was sacrificed to second by Asdrubal Cabrera, Sizemore walked. A groundout by Carroll advanced the runners, but Francisco grounded out.
Sizemore's seventh-inning double was a routine fly ball to left that was lost in the roof by Kubel, allowing Gutierrez to score. Francisco homered to lead off the eighth.
"We lost the game, so it doesn't really matter," Francisco said.
Tom Mastny threw one inning of scoreless relief, and former Twins setup man Juan Rincon pitched the eighth for Cleveland, entering to a smattering of boos offset by some applause. He promptly gave up a home run to Denard Span.
For the first time this year, Carroll, who has 57 starts at second base, started at third base.
Carroll had a pair of nice plays. He ranged in and to his left and, despite a double-pump after struggling to get the ball out of his glove, threw out Mike Redmond in the fourth. One inning later, he started an around-the-horn double play.
"I got a good view playing over there the other day at the end of the game, so it wasn't too bad," Carroll said.