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.
After A Rainy Sunday, A Return Of Winter On The Way
I will be sharing more specifics on the timing for this system in my forecast at 9, 9:30 and 10 p.m. on WBRC. Widespread showers are expected to return late tonight and into Monday morning ahead of our next cold front.
"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.
Kentucky governor 'guarantees' children sexually assaulted, poisoned due to school closures
The house voted 57-40 to override the veto of the tax increase and 66-28 to override the veto of the budget. You can stand here all day and act like you are all for ( education ) until it comes time to pay for it.
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".
Naldo screamer seals Schalke win over rivals Dortmund
Victory saw Schalke pull four points clear of third place, while Dortmund drop to fourth. Marco Pjaca sent the ball to Caligiuri, who stopped it for Naldo.
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.