Lester (13-6) give up just one hit before the seventh, when the Royals scored their only run to knock him out of the game with no outs. The lefty struck out three, walked four.
The Red Sox put the game away with a three-run eighth, sparked by Marco Scutaro’s double and back-to-back singles by Darnell McDonald and Adrian Gonzalez with no outs.
Danny Duffy (3-8) suffered his fourth straight loss after giving up two runs on five hits over six innings.
The Royals are now 4-13 over their last 17 games. They now play 16 of their next 19 on the road beginning Tuesday night in Toronto.