Moore (8-0) gave up one run, struck out three and walked one to become the first eight-game winner in the majors. The left-hander has won nine straight decisions, breaking the franchise record of eight set last year by Price.
Not only that, but the 23-year-old is the youngest lefty to start the season 8-0 since Ruth did it in 1917 at age 22.
"Historically speaking, it's been wonderful," Rays manager Joe Maddon said of Moore's season.
Moore is delighted with the results, especially because Tampa Bay is 9-0 in his nine starts.
"It's good for our team. That's the biggest thing we can take away from my record," he said. "On my days to pitch, we're winning all the games. That's the most important part. "
The Rays' ninth victory in 11 games left them a season-high three games over .500 (23-20) and lifted Tampa Bay into a third-place tie with skidding Baltimore in the AL East. The Rays had been in fourth since April 20.
"It puts us back in with a tie, standings-wise, which I think is important," Maddon said. "You got to get over each team that is in front of you before you can get back to the top. To come in here (and get a sweep) indicates how much better we've been playing."
Adam Jones had two hits, stole two bases and drove in a run for the Orioles, who have dropped five straight -- all at home. Baltimore managed only five hits, none after the sixth inning.
"He's dealing right now," Jones said of Moore.
Baltimore was outscored 22-16 in the first two games of the series. This time, poor pitching could not be blamed for the defeat.
"Pitching wasn't the issue today," manager Buck Showalter said. "It was trying to solve Moore. He was the difference-maker."
Chris Tillman (3-2) allowed three runs and five hits in six innings. He's 1-5 lifetime against the Rays.
Tampa Bay led 2-1 in the sixth inning when Joyce hit a drive that appeared to bounce off the wall near the right-field foul pole. First base umpire Dan Iassogna called it a fair ball but not a home run.
Showalter argued that it was a foul ball and Maddon contended it was a home run.
"It's kind of unusual for us both to be out there at the same time," Maddon said. "I had to argue the point that it could be a home run."
After a lengthy meeting, the umpires adjourned to watch a replay and returned with their decision: home run.
"I've got a guy who lets me know if I have a good argument, so I knew once they went inside it was going to be a home run," Showalter said.
In the bottom half, Danny Valencia hit a leadoff double in his first at-bat with Baltimore but was stranded.
One inning later, Manny Machado was credited with a triple after Jennings and Joyce nearly collided on a line drive to right that ticked off Joyce's glove and rolled to the wall. Jones followed with a run-scoring single, then stole two bases but was left at third.
Scott, who played for Baltimore from 2008-11, connected in the fifth for a 2-1 lead. He has at least one RBI in nine of 13 starts this season.
Tampa Bay's Evan Longoria extended his hitting streak to a career-high 13 games. ... Tampa Bay has held the lead in 30 of its past 31 games. ... The Rays recalled LHP Jeff Beliveau from Triple-A Durham and optioned LHP Alex Torres to the same club. Beliveau was sent back to Durham to make room for RHP Jake Odorizzi, who will be recalled Monday to start an afternoon game in Toronto. It will be his debut with the Rays. ... The Orioles recalled Valencia from Triple-A Norfolk after sending 2B Ryan Flaherty to Norfolk after Saturday's game. ... Baltimore opens a three-game series Monday night against the AL East-leading New York Yankees. Freddy Garcia will start for the Orioles. ... Jones has been successful on all seven stolen base attempts this season.