Top seed Caroline Wozniacki eased through to the third round of the Madrid Open with a 6-4 6-4 win over Serbia's Bojana Jovanovski.
In the men's first round Juan Martin del Potro overcame a hip injury to beat Mikhail Youzhny 6-1 3-6 6-3.
Argentina's Del Potro, who was out for eight months last year following wrist surgery, said he may not be fit for his round-two tie with Marin Cilic.
"I will check with the doctor if it's safe to play," said Del Potro.
The former US Open champion said the problem had first come to light in Portugal last week, where he won the Estoril Open.
"I started to feel it in the semi-final in Estoril last week but I really want to play this tournament," he said. "I will be careful for my future - I am playing really well last week and this, but my first goal is to play healthy.
"I want to be safe this week so I don't want to risk my presentation in French Open."
Talking about his return to fitness after last year's long lay-off, Del Potro admitted he had not expected to hit form so quickly.
"I'm surprised and I think people are by my level, but I know it's a long road to be 100% again. Hopefully in the second part of the year I'll be ready to fight for first positions."
After the match, world number one Wozniacki said on Twitter that it was "not the best game" but she was happy to get through.
She broke her opponent twice in the first set and three times in the second, while saving eight of the 11 break points she faced.
"Jovanovski is very young and I hadn't played her before," said the Dane. "She made some unforced errors, although later in the match she got better and didn't make it easy for me."
Elsewhere in the women's competition, third-seeded Francesca Schiavone progressed to the third round by beating fellow Italian Sara Errani 6-0 2-6 6-3.
The 2010 French Open finalist Samantha Stosur, the fifth seed, beat unseeded Slovakian Daniela Hantuchova 7-6 7-5, while Stosur's fellow Australian Jarmila Gajdosova came from a set down to beat 10th seed Agnieszka Radwanska 3-6 6-3 6-3.
One seed who made an early exit was number 11 Marion Bartoli, who went down 7-5 6-1 to Anastasia Pavlyuchenkova.
Britain's Andy Murray starts his singles campaign in the second round on Wednesday, when he faces Gilles Simon, who beat Ivan Ljubicic 7-5 7-6 (8-6).
While Murray had a first-round bye, those who won their way through to the second round included Marcel Granollers, who beat last week's Munich champion Nikolay Davydenko 7-5 6-2, and Germany's Florian Meyer, who defeated 16th seed Viktor Troicki 4-6 7-5 6-4.
Another seed who made an early exit was number 15 Fernando Verdasco, beaten 7-6 (9-7) 7-5 by Taiwan's Lu Yen-hsun.
In the second round, sixth seed David Ferrer beat Adrian Mannarino 7-5 0-6 6-0, while Argentine Juan Monaco knocked out ninth seed Gael Monfils after the Frenchman retired with the score at 6-2 3-0.
Monfils later revealed on Twitter that the problem arose because he had mistakenly eaten some cheese, which he is allergic to.
Afterwards Monfils revealed he had been forced out by an allergic reaction to some cheese he had eaten.
Murray will be hoping his singles campaign lasts longer than his involvement in the doubles. On Tuesday, Murray and brother Jamie lost 6-2 6-2 in the first round to Spain's Marc Lopez and Granollers.
Jamie Murray admitted they had been given "a bit of a dusting" but blamed their display on rustiness rather than injury.
"We struggled a lot," he said. "It wasn't the best match we ever played but for us it was a pretty difficult assignment straight off the bat.
"Neither of us had played matches for two or three weeks, maybe four weeks for me, so it was difficult."