The ninth-seeded Del Potro, a Roland Garros semi-finalist in 2009, received treatment on a sore left knee at the end of the second set before motoring through to prevail with a service winner on his first match point.
The 2009 US Open champion started brightly, sprinkling the court with forehand winners to open a 6-2 4-2 lead before seizing up as claycourt specialist Montanes fought back to take the second set tiebreak 7-5.
Del Potro, however, regrouped after the trainer strapped his left knee to set up a second-round meeting with Frenchman Edouard Roger Vasselin.
Del Potro, who won the Estoril Open on clay three weeks ago, said problems with his knee had started after his semi-final defeat at the Madrid Open two weeks ago.
"I'm still working with the physio every day, trying to recover," the 2009 US Open winner said.
"I'm glad to go through - I will have two days to work with the physio on my knee. So I have time to recover and to be in good shape for the next match."
The 26th-seeded Roddick, who has never gone past the thirdround of the Paris grand slam in 10 attempts, could not even blame his exit on being outfoxed by a claycourt specialist.
"It's not his favourite surface either," Roddick admitted during a news conference.
Instead he went down to the man who will long be remembered for his record 11-hour five-minute tussle with American John Isner on the green grass of Wimbledon in 2010.
Mahut fired 13 aces past former world number one Roddick to secure his place in the second round.
On Sunday, he spent only two hours and 22 minutes on Court Suzanne Lenglen, ending Roddick's cameo appearance with a backhand winner down the line.
The American, who had not played for two months between the Miami Masters and last week's World Team Championship in Duesseldorf because of a hip injury, refused to wallow in self pity.
"I'm not going to talk about this tonight, guys. I lost a match to a guy who played better than I did. We can ask it in as many different ways as we want," said the American.
"I'm going to not discuss it. I made a choice. I played. I'm fine. I lost.
"I move just horrendously out here. My first step is just so bad on this stuff. I feel like I'm always shuffling or hopping or not stopping or something. So my footwork on this stuff now is just really bad."
Local favourite Jo-Wilfried Tsonga overcame a sluggish start to advance into the second round with a 1-6 6-3 6-2 6-4 victory against unheralded Russian Andrey Kuznetsov.
World number 159 Kuznetsov, who had not played a single match on the main tour this year, caught the fifth-seeded Tsonga snoozing on Court Philippe Chatrier and bagged five games in a row to claim the opening set.
Tsonga, who last week said no Frenchman could win Roland Garros this year, made sure he would not endure a nightmare outing on the first day of his home slam and eventually crushed the 21-year-old Kuznetsov with some devastating forehand winners.
"Very often I have difficulties, I have to find my bearings," Tsonga told a news conference. "All players could tell you that practising has nothing to do with an official match. When you start a tournament, you don't have your bearings on the court."
He will next face German Cedrik-Marcel Stebe of Germany.
Former champion Juan Carlos Ferrero knocked out battling local player, and Roland Garros debutant, Jonathan Dasnieres de Veigy 6-1 6-4 6-3 in bright sunshine on the Suzanne Lenglen Court.
The 25-year-old Frenchman, cheered heartily by fans in the half-full stands, needed two bouts of treatment on his serving arm at changeovers and saved two match points before putting a forehand long on the third.
The Spaniard, who at 32 is one of the more seasoned campaigners in the men's draw, was not entirely happy with his performance but he said his age did not bother him.
"When I walk on the court I don't think whether I am young or old, the only thing I try to do is play well," the world number 44 and the champion in 2003 said.
Spaniard Ferrero will now play Marin Cilic after the Croat defeated Spain's Daniel Munoz 6-4 6-4 7-5.