Shane Victorino, Dustin Pedroia and Mike Napoli hit consecutive singles to start the third. Jimmy Rollins committed an error on a ground ball from Will Middlebrooks, which allowed two runs to score to make it 5-0.
Napoli later scored on a sacrifice fly from Bradley, giving Bradley four RBIs on the day.
Ben Revere drove in Humberto Quintero and later scored as the Phillies cut the deficit to 6-2 in the third. Revere was driven home twice by Chase Utley singles Sunday, the second time cutting the deficit to 6-3 in the fifth.
Utley then homered in the eighth off Red Sox righty Noe Ramirez to make it 6-5. Ryan Howard homered in the Phillies' next at-bat to tie the game.
But Shannon Wilkerson drove in Xander Bogaerts with a ninth-inning single to give Boston the lead for good.
Lee allowed nine hits and six runs (four earned) but was able to strike out seven with no walks in 5 1/3 innings. Red Sox right-hander Brandon Workman allowed three hits, two runs and two walks with one strikeout in three innings.
Red Sox center fielder Jacoby Ellsbury was replaced in the second after he jammed his right heel sliding back to first base on a pickoff attempt in the first. He will be re-evaluated Monday.
Up next: The Phillies take the short drive to Dunedin to play the Blue Jays in a Grapefruit League at 1:05 p.m. ET on Monday, available in an exclusive webcast. Phillies left-hander John Lannan makes his fifth Grapefruit League start. He is 1-1 with a 3.21 ERA. Relievers Zach Miner, Chad Durbin, Phillippe Aumont and Mike Adams are also scheduled to pitch. Phillies right-hander Kyle Kendrick is scheduled to pitch in a Minor League game.
Todd Zolecki is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.