The Indians scored twice in the top of the ninth, getting the go-ahead run on a wild pitch by closer Matt Lindstrom.
Axford came in seeking his fifth save in as many chances. He issued a leadoff walk to Jordan Danks, who stole second.
One out later, Ramirez homered over the left field fence.
The game was delayed 1 hour, 15 minutes at the start by the threat of rain. There was another 45-minute delay in the fourth.