Brandon Lowe had two hits, including a home run, and Yandy Diaz also went deep for the Rays.
Rays starter Jeffrey Springs was removed after three-plus innings due to ulnar neuritis in his left arm.
He struck out five and allowed one run on a homer by Red Sox outfielder Rob Refsnyder.
Nick Lodolo (2-0) worked around trouble in the first two innings, scattered eight hits and allowed two runs over five innings.
Toronto right-hander Chris Bassitt (1-2) allowed two runs, four hits and three walks while striking out seven in six innings.