The Finn was outside the top 100 after day one at Al Mouj Golf but a stunning 64 in round three put him alongside Stone in a six way tie for the lead with 18 holes to play.
Stone celebrated wildly as he holed a 20 footer on the last to get to 13 under but there was more drama to come as Välimäki holed from similar range for his own closing gain and a round of 70 to take the contest to extra holes.
The duo halved the 18th in pars twice but when Stone sent his second shot crashing into the stand on the third trip and failed to get up and down, a par was enough to hand Välimäki victory.
Välimäki had a stunning 2019 on the Pro Golf Tour, winning four times including three consecutive starts at the end of the season.
The 21-year-old claimed his card at the Qualifying School and after securing a first top ten at the ISPS Handa Vic Open, he is now a winner in just his sixth European Tour event.
"It's awesome," he said. "There are not many words to say, it's unbelievable.
"After the ninth hole when I made a double I knew that the tenth hole is good and I had to make a birdie on that.
"I didn't make a really good par on 11 and after that bogey I felt like, 'okay this is gone'. But then I just grinded, made three birdies and on the last, an awesome birdie."
Stone finished sixth at the Challenge Tour Grand Final at Al Mouj Golf in 2015 en route to graduating onto the European Tour and last season at this event he finished in the top 35 despite firing an 80 on Saturday.
He finished 103rd on last season's Race to Dubai Rankings Presented by Rolex as he had his first full European Tour season without a win but he is pleased with the work he has been doing despite missing out in Muscat.
“Obviously I’m a little disappointed but I’m really proud with how I played over the last four days,” he said. "It feels like I’m closer again. My confidence and swing are starting to come back. I’m feeling really comfortable and unfortunately I came up a little shy but the best thing about our job is I get to try again on Thursday.
"We’re professional sportsmen and full of emotion which showed with my putt on the 18th in regulation. I knew I needed to hole that to give myself a sniff. I managed to get it done. We leave it all out on the line, a lot of time and effort goes into this.
"Al Mouj Golf is one of the top five courses I’ve ever played on. Greg Norman has done an incredible job with the design and the groundstaff have done an amazing job in getting the course in the condition that it is. This is the first tournament I put on my calendar every season.”
Stone bogeyed the second but hit back with a birdie on the par five third to turn in level par as Välimäki enjoyed a roller coaster of a front nine.
He birdied the second from eight feet but dropped a shot on the fourth before making the most of the par five seventh and holing from 12 feet on the eighth to take the solo lead.
He got a bad lie in a bunker on the ninth and flew his third over the green as he recorded a double bogey but he hit back from ten feet on the tenth.
At that point he was in a share of the lead with Stone, who made a birdie from eight feet on the tenth, and two putts from just short of the green at the par five 12th meant it was the South African out in front.
Stone bogeyed the 13th, with Välimäki sandwiching a birdie on the 12th after damaging the cup with his approach with bogeys on the 11th and 14th.
Saddier had started with two bogeys but he birdied the fourth, sixth, seventh, eighth, 15th and 16th to lead alone before a bogey on the next put him alongside Stone.
Välimäki then got up and down on the par five 16th to also sit at 12 under before the late drama unfolded.
Italian Guido Migliozzi and Finn Mikko Korhonen finished at 11 under, two shots clear of Spaniard Alejandro Cañizares, South African George Coetzee, Danish teenager Rasmus Højgaard and England's Jordan Smith.