The Red Sox stripped theirs down by choice prior to the trade deadline, while the Yankees have been forced to restructure due to injuries.
Two of the beneficiaries of those circumstances will face off Saturday as the clubs continue a three-game series at Fenway Park with Boston's big return, slugger Yoenis Cespedes, likely to make his Red Sox debut.
Greene (2-1, 3.28 ERA) has been a bit shakier in his last two starts after arriving in the rotation with some impressive results. The right-hander has a 5.73 ERA and 0-1 record over his last two starts, allowing three runs and eight hits in 5 1-3 innings of Sunday's 5-4 loss to Toronto.
"I felt pretty good," Greene, who didn't receive a decision, told MLB's official website. "I made some good pitches. I've just got to be better later in the game."
Greene's first major league appearance came in Boston out of the bullpen. In one-third of an inning, he walked three and was charged with three unearned runs.
His two starts away from Yankee Stadium have been his best work with a 1.35 ERA.
Webster made his season debut Sunday after Boston shipped Jake Peavy to San Francisco. The right-hander limited Tampa Bay to two runs and three hits in 5 1-3 innings of a 3-2 victory for his second big league win.
"My body felt really good," Webster said. "My command wasn't as there as I wanted it to be, but I made the pitches when I needed."
He threw an inning of relief against the Yankees on Sept. 15 and surrendered a run, hit and walk.
Cespedes was acquired before Thursday's deadline in a deal that sent Jon Lester and Jonny Gomes to Oakland. He was held out of the lineup in Friday's 4-3 series-opening win.
The outfielder is a .313 hitter against the Yankees and .250 hitter at Fenway without a home run in 24 at-bats, but the Red Sox are hoping his right-handed bat and a 310-foot porch in left field results in plenty of power.
"I know I have some power," Cespedes said. "I can't predict how many home runs I'm going to hit, because when I'm out there I'm not looking to hit a home run, I'm just looking to make hard contact."
Without him, the Red Sox (49-60) won for just the second time in 10 games, over which they've been outscored 59-22.
Dustin Pedroia went 2 for 4 for a third straight game, and Will Middlebrooks returned for Boston after missing 67 games with a broken finger. The Red Sox could now be without catcher David Ross, who exited the game because of ongoing plantar fasciitis problems.
The Yankees (55-53) have lost five of six and fallen six games back of Baltimore in the AL East. Derek Jeter and Carlos Beltran homered in the opener.
"We've got to win games this time of year," said third baseman Chase Headley, who is 0 for 7 over two games after hitting safely in his first eight with the Yankees. "That's where the frustration comes."
While Beltran is hitting .448 with three homers and eight RBIs over an eight-game hitting streak, Jeter is batting .361 against the Red Sox in nine games this season and .373 over his last 15 at Fenway.