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Upton gave the Rays a 4-2 lead with a two-out hit off Jake Arrieta (2-7) in the fifth. Arrieta walked three during the inning.
"A clutch hit," Tampa Bay manager Joe Maddon said. "B.J.'s ball down the [left field] line is pretty large right there."
Moore (2-5) allowed two runs and seven hits over six innings. The left-hander had four losses in five starts since beating Seattle on May 1.
"I'm going to do everything I can to feed off this," Moore said. "It's good to get a win."
Moore has pitched with the lead in nine of his 11 starts this season.
"Matt really held it together well," Maddon said.
The AL East-leading Rays (31-23), with nine players on the disabled list, took two of three from Baltimore and lead the Orioles by one game.
"We've been able to hold serve," Maddon said. "We're still on top. The guys who have been out there have been doing a great job. The effort has been fantastic."
"Certain parts of the order, you'd really like to see him attack a little more," Baltimore manager Buck Showalter said of Arrieta. "I know it's frustrating for him, you see how good he's capable of being through spurts. But you're always going to have one inning, and it seemed like he couldn't get through it."
Arrieta threw just 41 of 83 pitches for strikes.
"I'm not pitching well right now, there's no doubt about that," Arrieta said.
The Rays extended the lead to 8-2 during a four-run sixth, aided by a pair of Baltimore errors.
Wieters made an errant throw from behind the plate attempting to pick off Ben Zobrist at third. After Elliot Johnson hit an RBI single, another run scored when third baseman Steve Tolleson misplayed Jose Molina's grounder. Carlos Pena added a sacrifice fly.
Tampa Bay took a 2-1 lead in the fourth when Matt Joyce drove in Pena with a sacrifice fly. Upton also scored from second on the play when Baltimore center fielder Adam Jones made a bad throw attempting to get Upton out at third.
Nick Johnson pinch hit for Jones, who was hit on the left wrist by a pitched ball Wednesday, in the ninth.
"I didn't want to take a chance, especially as banged up as we are," Showalter said.
Reynolds tied it at 2 on an RBI single in the fifth. He also had an eighth-inning solo shot.
Wieters put the Orioles up 1-0 on a second-inning leadoff homer. Baltimore then failed to score any additional runs despite having two on with no outs. Steve Pearce hit a flyball and both Tolleson and Robert Andino struck out.
Wieters also had a run-scoring hit in the ninth.
- Orioles INF Wilson Betemit was out of the lineup because of a fever.
- Tampa Bay RHP James Shields (6-3) will pitch against New York LHP Andy Pettitte (2-2) in the opener of a three-game series Tuesday night at Yankee Stadium. Shields is 0-4 with a 4.18 ERA in his last five road games against the Yankees.
- Baltimore RHP Jason Hammel (6-2) and Boston LHP Jon Lester (3-4) are scheduled starters for Tuesday night's game at Fenway Park.
- The Orioles purchased the contract of RHP Steve Johnson and optioned him to Triple-A Norfolk. To make room for Johnson on the 40-man roster, LHP Zach Britton (left shoulder) was transferred from the 15-day disabled list to the 60-day one.