Veerman produced a stunning 68 at Albatross Golf Resort on Sunday to move to 15 under par and claim a two shot victory in his 59th European Tour start.
Pulkkanen and Sean Crocker finished in a tie for second on 13 under, with Major Champion Stenson and Paul Peterson - who won here in 2016 - another stroke back.
Veerman had to remain patient in the early stages of his final round after watching overnight leader Pulkkanen pull four shots clear at the top of the leaderboard with two early birdies.
But he managed to do just that, following up a bogey on the second with birdies at the fourth, sixth and ninth to close the gap to one at the turn.
And when he holed his seven foot birdie putt at the tenth and Pulkkanen dropped a shot at the same hole, the American leapfrogged his playing partner into first place.
However, Veerman was not alone at the summit for long as he missed his par putt from six feet on the 11th to slip back to 13 under and find himself in a four-way share of the lead.
The lead changed hands again soon after as Crocker fired four consecutive birdies at the ninth, tenth, 11th and 12th to move to 14 under.
Veerman and Pulkkanen both made tap-in birdies at the 12th to join Crocker at the top once more.
Then Stenson holed a monster putt from 63 feet on the 13th green to join the co-leaders before a stunning approach from Pulkkanen at the 15th set up a close range birdie which would move him back into the outright lead.
After watching Pulkkanen knock in his birdie putt at the short 16th to move to 16 under, Veerman followed him in to keep the gap at one shot, while Stenson made a double bogey to see his challenge falter.
But a bogey from the Finn at the 17th meant there was nothing to separate Veerman and Pulkkanen heading to the last.
And as Pulkkanen found the rough and then the water at the 18th to surrender a double bogey, Veerman finished with a calm par to win by two shots.
Newly crowned champion Veerman said: “It feels amazing. I was asked if I could put it into words and I can’t.
“When I walked off the green I felt relief more than anything. Being neck-and-neck with Henrik, Tapio and Sean Crocker and a whole bunch of guys - to come out on top, breathe a sigh of relief. I’m elated. I feel amazing.
“It did (feel like match play). When Tapio made birdie on the 16th I knew I needed to make mine to stay within distance.
“Then on the 18th, I knew if I put one in play, it’s a simple hole, but it would put pressure on him because it’s not easy from the rough.
“To finally break out and win on the European Tour, it’s a dream come true being a European Tour winner.
“It’s something you practise for when you’re 12/13 years old, imagining making that putt. My putt was only a foot, so I wasn’t so nervous over it. I’ve got to soak it in. I feel amazing.”
Meanwhile, home favourite Ales Korinek gave the locals something extra special to cheer about as he fired a brilliant hole-in-one at the 208 yard 13th.