Dunne led by one shot overnight and a birdie on the 18th gave him a final-round 71 and outright second place.
Dunne got off to a nervous start quickly finding himself two behind Rahm who birdied the opening two holes while the diminutive Irishman was three behind when he bogeyed the fourth.
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The weekend's airstrikes in Syria and the country's seven-year-long civil war were also expected to be discussed. He said the strikes were unlikely to curb the government's ability to produce or launch new attacks.
"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 told reporters.
Rahm began the day third on the leaderboard, two shots behind Dunne, who had set the pace in the first three rounds, and one behind Elvira.
"It's hard to describe the pride I have right now to be Spanish national champion having been a Spanish amateur champion many times with the Spanish Golf Federation, and to be able to win as a pro and do this for them and for the Spanish people, it feels great and hard to describe", he said to Sky's Andrew Coltart.
The tournament was eventually decided on the short 17th as Rahm escaped with a par after nearly sending his tee shot into the water, the ball just holding up in the rough.
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Spaniard Elvira had seemed well placed to challenge Rahm, but he sent into his tee shot on the short 17th into the water and paid the price with a double bogey.
Further back, George Coetzee enjoyed an impressive final round as he ended the day on nine under, seeing him claim fourth place ahead of four others tied for fifth.
"It's been incredible", continued Rahm.
"It's probably the hardest Sunday I've ever had to play because the amount of crowds I was carrying".
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This isn't the first time that many of Trump's anti-interventionist supporters have felt betrayed by his foreign policy actions. He called for a strategy that leads to a political solution to "save it from the brutality of the Syrian regime".
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.