Baltimore led all three games entering the seventh.
Hill forced extra innings with an RBI single in the ninth off Jim Johnson in the Orioles closer's ninth blown save of the season and second in as many days.
Bud Norris (8-10), who started and threw five innings in San Francisco on Sunday, came on in the 14th and walked two batters ahead of Hill's single to take the loss.
Heath Bell (4-1) threw an inning to get the victory.
Hill followed Adam Eaton and Paul Goldschmidt as the Arizona last-inning stars.
Eaton's home run in the ninth gave Arizona the victory in the opener, then Goldschmidt homered in the ninth to tie it and hit another in the 11th to win it on Tuesday night.
Hill's game-winning single was the only hit in the five extra innings for the Diamondbacks, who finished the homestand 6-2. Baltimore went 4-4 in an eight-game trip to three NL West cities.
Norris, the Orioles' eighth pitcher, walked Parra and Martin Prado on four pitches apiece to start the 14th. He struck out Goldschmidt but Hill's single just got through the infield as Parra raced home with the winner.
The Orioles loaded the bases with one out in the 10th but Brad Ziegler induced Nate McLouth into a double-play grounder to second.
Baltimore took a 4-2 lead into the seventh on Monday and a 3-0 lead after six on Tuesday. The Orioles were up 4-2 again when Chris Tillman, bidding for his 15th victory, left the game.
Arizona got a run in the seventh on Prado's RBI single, then threatened but failed to score in the eighth.
Parra, who tripled in a run in the second, doubled to left off Johnson to start the ninth. Prado's bunt failed to advance the runner, then after a quick trip to the mound by manager Buck Showalter, Goldschmidt was walked intentionally.
That brought up Hill, who singled sharply to left on the first pitch to tie it. Goldschmidt moved to third on the play but was stranded when Matt Davidson struck out and Tuffy Gosewisch grounded to short.
All four Baltimore runs came with two outs against Patrick Corbin in the the second, two of them on Manny Machado's home run.
Taylor Teagarden and Brian Roberts singled with one out. They advanced to third and second on Tillman's groundout, then both scored on Nick Markakis' single. Machado's shot deep into the left field seats to made it 4-1.
Davidson, called by Arizona up from Triple-A Reno on Friday, got his first major league RBI with an infield single in the first. Didi Gregorius doubled and scored on Parra's two-out triple off the wall in center to cut the lead to 4-2 after two.
Corbin failed for the fourth time in his bid for a 13th victory. He went seven innings, giving up four runs on nine hits with seven strikeouts and one walk.
NOTES: When they recalled Campana, the Diamondbacks optioned LHP Tyler Skaggs to Triple-A Reno. Skaggs had been with the team for a couple of days as a long-inning relief option in his latest in a series of stints with the big league club. ... Three pitchers were used as pinch hitters, Randall Delgado and Wade Miley for Arizona, Scott Feldman for Baltimore. ... The Orioles return home for a three-game interleague series with Colorado, sending LHP Wei-Yin Chen (6-5, 3.06 ERA) to the mound against the Rockies' RHP Juan Nicasio (6-6, 5.04) in the opener Friday night. ... Arizona leaves on a 10-game road trip, beginning with a three-game series in Pittsburgh. RHP Brandon McCarthy (2-6, 4.73) goes for the Diamondbacks in Friday's opener with RHP Gerrit Cole (5-5, 3.95) starting for the Pirates.
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