He came home in 36 to allow Lorenzo-Vera back into the contest but a wonderful approach to eight feet set up a birdie on the first extra hole and handed him victory.
Lagergren's closing 68 got him to 16 under alongside Lorenzo-Vera - who signed for a 70 - and one shot clear of England's Andy Sullivan and Australian Lucas Herbert, who matched the lowest round of the week with a 63.
Dane Lucas Bjerregaard was then at 14 under, one shot ahead of Frenchman Julien Guerrier.
Lagergren's triumph makes him the second first-time Swedish winner in three weeks after Alexander Björk's victory at the Volvo China Open, and comes in his 131st European Tour appearance.
"I've been waiting for this moment my whole career," he said. "I'm so relieved and I'm so happy at the moment, I can't describe it in words. I've just been fighting so hard to get to here, it's just awesome.
"When you're just missing out a few times you really want to get that first win and it feels like you really need to get over that first step to then really continue and get more and more. This is a big step for me."
For Lorenzo-Vera, it is a second consecutive second-placed finish after he lost in the final at last week's GolfSixes alongside Romain Wattel.
"What can I say? He made a birdie on the last and I didn't really have a chance," said an emotional Lorenzo-Vera. "It was the kind of day where I was a bit far from the pins so it was hard to score low for me. He took his chance better than me.
"I gave it a go, I tried my best, he did it better than me. Now I'm just going to have to work more to be able to finish the job."
Lagergren holed a brilliant 30-footer for his opening birdie, with Lorenzo-Vera taking advantage of a lovely approach to six feet before giving the shot straight back on the next.
The 26 year old Lagergren holed another long putt on the difficult sixth to take a share of top spot and then hit the front on his own with a tee-shot to 12 feet on the par three seventh.
Sullivan birdied the third but, after a poor second on the sixth, he did very well to drop just a single shot and that appeared to spur him on. The 30 year old holed a long putt on the seventh and a six-footer on the eighth and, when he put a stunning second to six feet for an eagle on the ninth, he was just three off the lead.
Lagergren then made a two-putt birdie of his own on the same hole to lead by two at the turn but Sullivan took advantage of the par five 12th to keep in touch with the leader.
Lorenzo-Vera broke a run of ten pars with a wonderful tee-shot to three feet on the 13th but with both he and Lagergren failing to get up and down from the sand on the next, the door was open for Sullivan.
The three-time European Tour winner spun his approach to the penultimate hole to three feet to move into second and he was soon joined there by Herbert.
The 22 year old had started the day eight shots off the lead but after a birdie-bogey start he added gains on the fourth, eighth, ninth, tenth, 12th, 15th, 17th and last to set the clubhouse target just one off the lead, where Sullivan joined him after a par on the last.
Lorenzo-Vera found a nice lie after a poor tee-shot on the 17th and made the most of it, putting his approach to three feet to share the lead heading up the last.
Bjerregaard overcame an opening bogey with gains on the fourth, ninth, tenth and 12th to be right in the mix and missed a host of chances on the back nine before finding a bush on the last for a closing bogey.
Guerrier was four over for the day after five holes but fought back with birdies on the ninth, tenth, 12th, 15th and 16th before he too bogeyed the last.
England's Steven Brown and home favourite Francesco Laporta were at 11 under, a shot clear of two more Italians in Edoardo Molinari and Andrea Pavan.
Simon Khan made the 16th hole-in-one of the season and the second in as many days on the 15th when he holed a seven iron from 182 yards.