Evan Longoria followed Sizemore's home run with a single and reached second base on an error, but Anderson retired the last five batters he faced. The 24 year-old helped Cleveland snap a three-game losing streak.
Yan Gomes had three hits, including an eighth-inning homer, while Michael Brantley added three hits to lead Cleveland, who was just shut out twice on Sunday in a doubleheader in Baltimore.
Anderson gave up just six hits over 7 2/3 innings against the Rays in his big league debut in Cleveland on June 21st. He struck out two and did not walk a batter while throwing 100 pitches.
Monday was the second time in less than a week that a pitcher flirted with perfection against the Rays. Marco Estrada of the Toronto Blue Jays took his bid into the eighth inning on June 24th before Logan Forsythe broke it up with an infield single. The Blue Jays won the game 1-0 in 12 innings.
Nathan Karns suffered the loss for the Rays to drop his record to 4-4. He was also on the mound during Estrada's perfect game bid against Tampa Bay, who has lost five of their last seven games. Karns didn't allow a hit until the sixth inning against Toronto.
The Rays' rookie allowed two runs on nine hits in six innings against the Indians.
Brantley knocked home Jason Kipnis with a single in the first inning to help the Indians end a 19-inning scoreless streak. Kipnis knocked in the second run with a base hit in the second inning.
The Indians and Rays continue their series at Tropicana Field tonight.