Eovaldi had been scheduled to start the Rays' game against the New York Yankees on Wednesday but was scratched about two hours before game time.
"I assumed there was a possibility, of course, especially where I am in my career and the way my contract was for this year," Eovaldi said before leaving Tropicana Field. "I was assuming that I would be traded, but as for what team it would be, I wasn't thinking about that at all."
He'll make his Red Sox debut Sunday.
Eovaldi can become a free agent after this season.
The deal was announced less than 12 hours after Boston's main American League East rival, the Yankees, obtained reliever Zach Britton from Baltimore.
Eovaldi, 28, returned this season following Tommy John surgery performed in August 2016, going 3-4 with a 4.26 ERA in 10 starts. He started 2018 on the disabled list after undergoing arthroscopic surgery on the elbow at the end of spring training.
He is well aware of the New York-Boston rivalry, going 23-11 with a 4.45 ERA in 51 games while playing for the Yankees in 2015-16.
"Those games are definitely exciting," Eovaldi said. "The fans are ... it's kind of one of those indescribable tension is always high and it's always a good series."
Eovaldi has averaged 96.9 mph on his fastball this season, according to Statcast. He has allowed more homers (11) than walks (8), and has 53 strikeouts over 57 innings.
Beeks, 25, made the first two major league appearances of his career this season, allowing nine runs on 11 hits in 6⅓ innings for a 12.79 ERA. He is 5-5 with a 2.89 ERA in 16 starts for Triple-A Pawtucket.
The division-leading Red Sox entered Wednesday five games ahead of the Yankees.
The Associated Press contributed to this report.