Aaron Hicks drove in two runs without a hit and Jacoby Ellsbury added an RBI single.
Snell agreed with his skipper that he "yanked" his final pitch, a 3-2 slider to Hicks and that his workload in the fifth inning exhausted him out.
A two-run homer by Judge off Ramirez extended the lead to 8-3.
Xavier Cedeno fielded the ball and turned toward second, hoping to start an inning-ending double play but forgetting the infield was in and no one was covering the base.
After Judge's single against Diaz tied the score 3-3 in the sixth, the Yankees had runners at the corners and one out when Gardner batted for Pete Kozma and hit a comebacker. Gardner also left the game and the Yankees said he bruised his jaw and strained his neck. Cash told reporters Weeks avoided serious injury but was clear with reporters that Weeks's inexperience at the position (just three career starts a first) was not a factor. "For the first start I'm pretty pleased".
He allowed a long two-run homer to Rickie Weeks in the first inning after a two-out walk to Evan Longoria, then settled in.
On Wednesday afternoon, all eyes were on Jordan Montgomery as he made his Major League debut.
The Yankees weren't supposed to need a fifth starter until April 16, but the decision came a week early: lefty Jordan Montgomery has won the job, and manager Joe Girardi announced after Monday's game that Montgomery will make his major-league debut on Wednesday, not Saturday as originally planned.
Bryan Mitchell (1-0) relieved with a runner on second, and second baseman Starlin Castro allowed Weeks' hard grounder up the middle to bounce off his glove for an error that made it 3-0.
Montgomery threw 61 of 89 pitches for strikes, including 17 missed swings.
The Yankees will likely be shorthanded in their bid for a fourth straight win.
He started Tampa's season opener last Thursday, striking out eight while allowing three hits, one walk and one unearned run in five innings of a 2-1 loss to Lakeland.
Yankees: RHP James Kaprielian, the 16th overall draft pick in 2015, could be headed to Tommy John surgery after an evaluation by Dr. Neil ElAttrache. After being sidelined by a sore right ankle and a stomach ailment, he was benched against a left-handed starter.
Yankees flamethrower Luis Severino (0-0, 7.20 ERA) will oppose Rays' righty Matt Andriese (0-0, 9.00 ERA).