The top-seeded Czech broke for a 5-3 lead in the final set on a double fault from Monfils. He wrapped it up with his 12th ace.
"It was very close in the end. I'm happy that I was able to do it," Berdych told Canal Plus television. "Actually, it's even more special to beat the home player, unfortunately for the crowd."
Berdych relied on a strong serve and deep groundstrokes to grab the lead, but made several unforced errors to drop the second set.
"Tomas had a great match," Monfils said. "He got off to a good start, so I had less maneuvering room. I tried my best. I tried to return better, I tried to play with a bit more aggression too."
Berdych broke Monfils for a 2-0 lead when the Frenchman sent a backhand long. The 2010 Wimbledon finalist then saved three break points to hold serve before hitting two consecutive return winners to take the first set.
In the second set, Berdych double-faulted to give Monfils a break chance that the Frenchman converted for a 3-2 lead when the Czech's backhand volley sailed long.
Monfils erased three break points at 4-3 and leveled at one set each on a forehand error from Berdych.
Berdych also won their only previous meeting, four years ago in Dubai.