Playing at the season-ending tournament for the first time since 2007, Gasquet produced some stunning shots but struggled to return del Potro's big serve in a match laced with unforced errors from both sides.
Wawrinka, the only first-timer in this year's tournament, got off to a strong start and then recovered from a second-set tiebreaker in which he failed to win a point.
"My first impression was really, really nice to be here," the Swiss star said. "I saw that tournament so many times on TV. To win my first match here was something really special."
Wawrinka spent his career in the shadow of Federer before reaching the semifinals at the US Open this year. Berdych reached the semifinals of this tournament two years ago.
After the round-robin stage, the top two finishers in each group advance to the semifinals.
Wawrinka served well throughout the match, hitting 11 aces and winning 66 percent of points on his second serve. He also saved the five break points he faced.
Wawrinka broke in the sixth game of the first set when Berdych sent a backhand wide. He then opened up a 5-2 lead before earning two set points on his serve with a forehand winner. However, he missed three consecutive easy shots to offer Berdych his first break point of the match. But the Swiss raised his game at the right time and converted his fifth set point.
Berdych hit his racket against the floor in anger after wasting a break point by botching a forehand in the fourth game of the second set. He missed three other chances to break but sailed through the tiebreaker.
The Swiss recovered in the decider, breaking Berdych for a 3-1 lead after he double-faulted and sent a forehand long. Wawrinka, whose thigh was massaged in the deciding set, sealed the victory on his first match point when Berdych sent a forehand wide.
"As I saw the match, well, it was a very close game again," Berdych said. "It's been really about the small chances."
Information from The Associated Press was used in this report.