The Englishman's 72-hole total was 10-under 278. He started the round one-stroke back of compatriot Bradley Dredge, the overnight leader.
Finch bogeyed the fourth but then went out to a four-stroke lead with a six-foot birdie putt at No. 15 and held on to get his second win on the European Tour.
"It hasn't sunk in yet really," Finch said. "I'm delighted with how I played all day. I'm really pleased with how I was mentally. I tried to keep going forward this time. I was a lot more in control this time."
Aguilar, from Chile, finished at 8-under 280 after also shooting a closing 2-under 70.