Top-seeded Wozniacki, 20, took just 60 minutes to dispose of the 40-year-old Date-Krumm and will now face her near-namesake Aleksandra Wozniak of Canada for a place in the last 32.
"It was a good start to the tournament for me," said Wozniacki.
"The last time I played Kimiko it was a bit of a longer match, at Wimbledon (in 2009, when Wozniacki won 5-7, 6-3, 6-1).
"She plays a different kind of tennis so it was difficult to play as I wanted to. But a win is a win, so I'm happy to be through to the next round.
"Physically, I'm feeling good and ready to play this tournament."
The Danish star, who won her fourth title of 2011 in Brussels last week, is bidding to go beyond the French Open quarter-finals for the first time, having lost to eventual champion Francesca Schiavone in the last eight last year.
Clad in a blue dress and with heavy strapping on her left thigh, Wozniacki broke Date-Krumm at the first attempt on Court Philippe Chatrier and then saved break point with an ace before going 3-0 up.
An inopportune double-fault from Date-Krumm allowed Wozniacki to move 4-0 ahead and moments later the Dane took the set by charging to the net to gobble up an under-hit drop-shot from her opponent.
Date-Krumm, who made her Roland Garros debut a year before Wozniacki was born, was broken in the sixth game of the second set and wilted on the first match point when she sent a forehand beyond the baseline.
Earlier, defending champion Schiavone swept into the second round by beating American 19-year-old Melanie Oudin 6-2, 6-0 on centre court.
Fifth seed Schiavone, who last year became the first Italian woman to win a Grand Slam, will meet Russian Vesna Dolonts for a third-round berth.
"I'm still shaking a little bit," admitted Schiavone, after returning to the scene of her emotional victory over Australia's Samantha Stosur in last year's final. "There was a lot of adrenaline. I felt really happy to be there."
Russian third seed Vera Zvonareva, a losing finalist at both Wimbledon and the US Open in 2010, joined Schiavone in the second round when she overcame Spanish veteran Lourdes Dominguez Lino 6-3, 6-3 on Court Suzanne Lenglen.
Zvonareva's next opponent is German qualifier Sabine Lisicki.
Home favourite Marion Bartoli, the 11th seed, flirted with becoming the biggest casualty in the women's draw so far, conceding the first set to Georgian Grand Slam debutant Anna Tatishvili before storming back to win 1-6, 6-2, 6-1.
"It was an absolutely catastrophic start," said Bartoli.
"Even in my worst nightmares, I couldn't have imagined a start like that. But in the second set I started from zero, forgot about the first set, and it started to get better."
Czech ninth seed Petra Kvitova, who triumphed in the Madrid Masters on clay earlier this month, eased into the second round with a 6-2, 6-1 thrashing of Hungary's Greta Arn.
Agnieszka Radwanska, the 12th seed from Poland, encountered similarly limited resistance from Austria's Patricia Mayr-Achleitner, winning 6-1, 6-2.
There was no such joy, however, for Russian 26th seed Nadia Petrova and Czech number 31 seed Klara Zakopalova, who both went out.
Petrova went down 6-7 (5/7), 6-3, 6-4 to Australia's Anastasia Rodionova, with Zakopalova falling 7-5, 6-1 against Chan Yung-Jan of Taiwan.
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