"I guess that's pretty cool," said Betts, of setting a franchise record. "I'm just trying to do my job, get on base and score a run."
Betts, 24, had been locked in a tie with Jacoby Ellsbury for the franchise record in leadoff homers. Ellsbury played 715 games for the Red Sox from 2007-13. Betts broke the record in 441 games.
The blast against Archer was the third leadoff homer for Betts this season.
Ian Browne has covered the Red Sox for MLB.com since 2002. Follow him on Twitter @IanMBrowne and Facebook. This story was not subject to the approval of Major League Baseball or its clubs.