Defending champion Sharapova was nursing a cold but opened her bid for a third Roland Garros title in four years with a 6-2 6-4 victory over Estonian Kaia Kanepi on a breezy Court Phillipe Chatrier.
Men's third seed Murray then took his winning streak on the red dust to 11 matches by overpowering Argentine lucky loser Facundo Arguello 6-3 6-3 6-1, maintaining his unbeaten run since marrying long-term partner Kim Sears.
Sharapova, who was also champion in 2012 and runner-up in 2013 despite an intense dislike for the surface earlier in her illustrious career, produced her usual power tennis to ease through to the second round.
It was not a perfect performance, perhaps due to the cough that was troubling her, but she never looked in danger as she set up a meeting with fellow Russian Vitalia Diatchenko.
Sharapova left to a few boos though after declining an on-court interview.
"Unless I'm really in my coffin I'm going to be out there," she told reporters. "I got sick a week before the tournament, not right before.
"I guess it's a little bit better timing but it's just the way it is. I'm getting over it, hopefully it will pass soon."
DARK HORSE
The world number two is in the opposite side of the draw to her old nemesis, top seed Serena Williams, and will need to be 100 percent to negotiate her way through to the final.
Former runner-up Samantha Stosur, an easy winner against American Madison Brengle, could await in the third round while eighth seed Carla Suarez Navarro of Spain, tipped as a dark horse for the title, is a possible foe in the quarter-finals.
Suarez Navarro beat Monica Niculescu of Romania 6-2 6-2 but several seeds tumbled on day two in swanky western Paris.
Spain's Feliciano Lopez, the men's 11th seed, became the highest casualty so far, losing 6-3 7-6 (9) 6-3 to Russian Teymuraz Gabashvili while women's 14th seed Agnieszka Radwanska's slump continued as the former world number two succumbed 6-2 3-6 6-1 to Germany's Annika Beck.
American veteran Venus Williams, seeded 15, also lost, although a 7-6 (5) 6-1 defeat by compatriot Sloane Stephens barely registered on the shock scale.
With nine-times champion Rafa Nadal and top seed Novak Djokovic keeping their powder dry until Tuesday, Murray joined Roger Federer in getting a head start with a confident display on Court Phillipe Chatrier.
There were a few early nerves, a double-fault in the opening game giving Arguello a break point that he could not convert and again when he allowed his 137th-ranked opponent to recover from 3-0 down to level at 3-3 in the first set.
Once Murray, twice a French Open semi-finalist, broke in the eighth game though he was untroubled.
"I hope (nerves) are always there," he said. "I can't remember the last time I played a first-round match in a slam and did not feel nervous."
Murray was joined in the second round by fourth seed Tomas Berdych who beat Yoshihito Nishioka 6-0 7-5 6-3.
Nishioka was one of seven teenagers in the men's draw, two of whom produced excellent wins on Monday.
Croatian Borna Coric, 18, beat American Sam Querrey 7-6 (8) 6-3 0-6 6-3 while Australian wildcard Thanasi Kokkinakis, 19, put out Georgian Nikoloz Basilashvili.
Kokkinakis will face countryman Bernard Tomic next.
Nick Kyrgios, yet another Aussie young gun, lived up to his 29th seeding by breezing past Denis Istomin.
Home crowds flocking in to the grounds had plenty to cheer too with French number one Alize Cornet opening proceedings on centre court with a fighting 4-6 6-4 6-1 triumph over Italian Roberta Vinci and the ever-popular Gael Monfils, the 13th seed, defeating compatriot Edouard Roger-Vasselin.
