Dunne, the British Masters champion, was "disappointed" to miss out on a second European Tour victory but, at the same time, expressed his delight for the victor.
With AP Photos.Jon Rahm shot a 5-under 67 to win the Spanish Open with overnight leader Paul Dunne two shots behind. They were pushing so hard and I felt that. Some of them skidded, some of them bounced soft. When I played for one, I got the other result. "I was hitting it long and straight, so I feel good going into the next few weeks".
"I've been blessed to be national champion from 16 to all ages in Spain".
Rahm chipped in for birdie on the 10th to take the lead for the third time in the round and then was two clear after 13.
Two ahead playing the par-five 18th, Rahm reduced it to a drive, a mid-iron and two putts on greens that were far from ideal.
"It's hard to describe the feeling of satisfaction and pride right now", Rahm said. "The feeling of pride and satisfaction is awesome". Everyone seemed to want me to win. I felt that tension. I tried to isolate myself, but I could feel it. Dunne picked up $206,000, and Elvira won $116,000.
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He had been hunting a first title as an expert past year, to grow his two previous European Tour wins in Dublin and Dubai.
But this was an even more impressive win considering the pressure on his shoulders. He needed to go super-low on Sunday to have any hope of snatching his first European Tour win and set out in ideal fashion with a trio of opening birdies.
Elvira holed a birdie from a similar distance at the same hole and moved alongside Rahm by draining a 35-footer at the 14th, where Dunne slipped three behind with a poor chip and bogey-five.
In all, he made 21 birdies across his four rounds, two eagles and only had 3 bogeys and a single double bogey on the par four 12th on day.
While he birdied the 15th to get back to within two shots of Rahm and Elvira, there were to be no late mistakes from the eventual victor.