Del Potro had suffered a distressing journey to London after being robbed at Paris's Gare du Nord railway station, an incident which cost him his passport and a rosary blessed by Pope Francis.
The former US Open champion, who lost to Roger Federer in the quarter-finals of the Paris Masters on Friday, spent two hours talking to French police and sorting out paperwork before boarding the Eurostar and heading to the O2 Arena.
But the painful trip has proved worthwhile for fourth seed del Potro, who finally subdued a lively challenge from 27-year-old Gasquet in two hours and 23 minutes in the opening Group B clash.
The victory was crucial for del Potro, who faces Novak Djokovic and Roger Federer in his remaining group matches.
"I'm so happy to qualify once again here. I have fantastic memories from Wimbledon and the Olympics and I always play well in London," Del Potro said.
"Richard has an unbelievable backhand and he is a good fighter, but in the end I played solid and won.
"Hopefully I can beat Novak and Roger. It's going to be really tough, but I don't have anything to lose against them."
Del Potro has claimed four titles this year and came desperately close to reaching the Wimbledon final, only for Djokovic to beat him in five sets in the semi-finals.
The 25-year-old arrived in London in good form, with two of his tournament victories coming in the last month in Tokyo and Basel, while he also beat Rafael Nadal to make the final of the Shanghai Masters.
Gasquet was making his first appearance at the Tour Finals since 2007 after rising to ninth in the world rankings on the back of one of the better seasons in his mercurial career.
With Wimbledon champion Andy Murray sidelined due to a back injury, Gasquet's ranking was high enough to earn entry into a tournament usually reserved for the world's top eight players.
Possessing one of the most sublime one-handed backhands in the game has made Gasquet something of a treasure for tennis purists, but his talent has never quite produced the results his admirers hoped for until this year, when he made the semi-finals of the US Open.
Fittingly for two in-form players, this was a high-quality encounter right from the opening 32-shot rally.
Gasquet looked to have seized the momentum when he broke for a 5-3 lead in the first set, but del Potro broke back as the Frenchman served for the set.
It needed a tie-break to settle the set and Gasquet took it, unloading one of his majestic backhands to secure a mini-break before a forehand clipped the net fortuitously to give the eighth seed a one-set lead.
Del Potro has gone the distance plenty of time in the past, however, and he broke in the fourth game of the second set before easily levelling the match.
That set up a tense deciding set which seemed to have swung del Potro's way when he broke for a 4-3 lead.
Gasquet then broke del Potro as he served for the match, but the Argentine hit straight back with another break before finally closing out the victory.