The U.S Open champion produced a disciplined display on the slow Roland Garros clay to dismiss his 33-year-old opponent with some ease on centre court.
The match ended in minor controversy, however, with Martin initially failing to shake Roddick's hand after a disputed line call in the third set.
Despite his impressive victory Roddick, the world's fastest server, was quick to play down his chances of winning the title on a surface that ill suits his power-based game. The 21-year-old has never been beyond the third round in Paris and suffered upset first-round defeats in 2002 and 2003.
"It's good to get a match under my belt here and break my jinx here," Roddick said. "It would not have been fun to get a Triple Crown here.
"My goal is to make the second week."
Roddick looked pale and said he was suffering from an upset stomach, echoing a complaint by British ninth seed Tim Henman who also said he felt unwell despite winning his first round match on Monday.
"I guess there's a one or two-day virus going around Paris at the moment," Roddick said. "I got it last night and hopefully I will wake up tomorrow and have a bit of an appetite."
He sneaked the first set by recovering from 4-2 down to win the tiebreak 7-5 with a crunching forehand and took control of the second set with an early break.
The two players were involved in an acrimonious exchange after Martin returned a Roddick shot, then called it out and stopped playing to win a point with the score 4-3 in his favour in the third set.
Roddick angrily questioned the decision but still held serve and forced successive errors from Martin at 5-5 to break decisively before wrapping up a two-hour victory to set up a second-round meeting with Frenchman Olivier Mutis.
The argument soured the end of the match, although both players later played it down.
"I have nothing but respect for Todd," Roddick said. "I guess he thought I was trying to question his call there at the end but I didn't mean any disrespect at all. I was just questioning the ruling of the umpire."
Martin, who admitted he may not play at Roland Garros again, tried to make light of the incident.
"I made a royal you-know-what out of myself," he said. "But Andy and I spoke in the locker room. He didn't understand the rule and I believe him.
"For one momentÂ…it seemed to me he forgot who was on the other side of the net."
Martin showed he had not lost his sense of humour when he added: "It makes me feel pretty bad that I can't beat a guy who's throwing up."
