The 26-year-old Hidalgo had missed a short birdie putt at the same 18th hole to allow fellow Spaniard Rahm into the play-off, giving the two-time major champion a chance to become the first player to win four Spanish Open titles.
Rahm had birdied the final two holes of his round at the Club de Campo Villa de Madrid.
"Insane, absolutely insane," Hidalgo said after Sunday's triumph. "It was weird because I was pretty relaxed all day, even when I missed this short putt on hole 72."
After Hidalgo pushed his putt wide, the players were back on the 18th for the first playoff hole. Rahm went over the green with his drive, chipped close and then rolled in his birdie putt. Hidalgo did not miss his short attempt for birdie this time.
Back on the 18th, Hidalgo just missed the green and Rahm went left. Rahm had a poor chip and went over the green, while Hidalgo put it close again and drained the putt to take his first victory.
Hidalgo, who had never finished in the top three in 79 previous European tour events, came into the final round with a two-shot lead over Rahm. Hidalgo shot a one-under 70 after making three bogeys in his first six holes for a 14-under 270 overall.
Rahm, the former No.1, finished with a three-under 68 and seven birdies, including three of his final five holes. He also had two bogeys and a double bogey.
Rahm had trailed by five shots heading into Saturday. He arrived late in the Spanish capital after the birth of his third child.
Rahm won the LIV individual title and was back in Europe to try to reach the minimum of four European Tour starts required to be considered for the Ryder Cup next year. He is being allowed to play after appealing the European tour sanctions against him for playing the LIV circuit.
Tommy Fleetwood shot a five-under 66 for his best round of the week to finish in a tie for third.
Patrick Reed ended in a tie for 10th after an even-par 71 that included two double bogeys. Tyrrell Hatton also was 10th after a three-under 68. Shane Lowry was tied for 13th after a two-under 69 that also included a double bogey.
Australia's David Micheluzzi also shot a 71, finishing the weekend two-under and in joint 39th place. Compatriot Jason Scrivener was equal 57th after also finishing with a 71.
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