The Latvian triumphed 6-1, 7-6 (5) for his sixth career tournament victory.
Gulbis secured the opening set in just 26 minutes, but then wasted three break points early in the second. He fell behind in the tiebreaker, but fought back to win five of the next six points and secured the victory when Delbonis hit a forehand into the net.
''I am really happy to win here,'' said Gulbis, who should rise to a career-high of 15 in the rankings. ''My next goal is to be top 10, even top eight. I think if I play well in the Grand Slams, I have a good chance. It's tough to kick out somebody from the top as there's a lot of good players but I am working on it.''
Gulbis dropped just three points on his first serve - and none in the first set - and also hit nine aces.
The seventh-seeded Delbonis also wasted two break points in the second set.
''He played well, I didn't start the best way and he broke me twice,'' Delbonis said. ''In the second set I gave my best, it was more equal. I had my chances but couldn't take them because he served well. I'm happy for this week and although this match wasn't the best, I'm happy for playing it.''
Gulbis faces Lukasz Kubot in the first round of the French Open, while Delbonis will play Dusan Lajovic.