Lowry started the round four shots off the lead, but he made an eagle on the par-4 11th to move ahead of overnight leader Bernd Wiesberger.
The 25-year-old Lowry then traded the lead with Fisher before taking advantage of Fisher's bogey on the 16th with a birdie on the 17th.
Fisher had one last chance after Lowry bogeyed the 18th, but the Englishman also had a bogey to allow Lowry to claim his second career European Tour win.
"I cannot believe this," Lowry said. "I can't explain how I feel, really. It's a dream come true; I'm over the moon. I thought this could be my day, and thankfully it is."
Lowry shot 14 under through the four rounds on the Oceanico Victoria Golf Course. He won the Irish Open three years ago as an amateur.
"It was like playing in Ireland. It was crazy; it was great," he said. "Everyone was referring to me as the fellow who won the Irish Open as an amateur, but now I've won such a prestigious event I don't know what to say."