Rahm, Dunne and Elvira were tied for the lead on two occasions, but it was the 23-year-old who held his nerve in his 19th Tour event to become the third-quickest player to three titles.
"It's been wonderful. It's truly been the hardest Sunday I've ever had in any tournament that I've won because the crowd wanted it so much and I wanted it so much".
"This is hard to describe", said an emotional Rahm at the finish.
"When I made the decision to come straight from Augusta it wouldn't be to just show up and walk around, I wanted to win this tournament", Rahm said.
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"I will keep doing what I am doing - get through this evening because I am a little bit disappointed - but Spain has a great champion in Jon and I am sure he will win many more of these". "To round my amateur and pro career together in this way and win the last one I had to win and join that prestigious list of Spanish winners, it's hard to explain how good it feels and how satisfying it is".
Rahm played a lovely chip at the first to set up a birdie and when he hit a huge drive on the second to set up another gain, he was in a share of the lead with Dunne.
Elvira carded a 66 to stand at one shot behind while Rahm is joint-third with Sweden's Henric Sturehed, both of whom matched Elvira's six under for the day.
It gave him a two-shot victory over Ireland's Paul Dunne.
After Rahm had stamped his mark on proceedings with birdies at the first, second and seventh holes in the final round, the tournament was decided on the short 17th.
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Rahm again incurred the wrath of Sky commentators, including renowned Dougie Donnelly, for a second day running, in failing to call "Fore" after a wild drive off the 10th that would lead to a chip-in birdie and take him to 18-under.
Rahm then birdied to one clear on 18-under but dropped a shot on nine with Dunne parring four holes in succession from the sixth with the duo heading to the closing nine locked at 17-under.
Dunne bogeyed 14 but regained the shot with a birdie on 15 before securing second on his own with a 72nd hole birdie and a 23rd birdie over the four days.
Coetzee equalled the lowest round of the week with a 63.
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I do not understand how we will misunderstand his intent. And to the times when I don't do it well, that's on me. So that just kind of makes me angry'.