Price allowed two hits -- singles in the fifth and sixth innings -- in collecting his major-league high 13th victory against four losses. He walked three and struck out seven.
Davis faced just six batters over two innings, striking out in the side in the ninth.
Luke Scott opened the scoring in the third inning with his 12th home run of the season.
The big Tampa Bay rally started when Jimenez hit Desmond Jennings with a pitch with two outs in the sixth. Jose Morales followed with a single and Sean Reynolds walked to load the bases.
Upton then jerked a pitch down into the left-field corner for a double that made the score 4-0. He reached home when Carlos Pena greeted reliever Scott Barnes with an RBI single.
Back-to-back doubles from Scott and Jeff Keppinger in the seventh completed the scoring.