Kvitova, seeded sixth, broke the Slovak qualifier five times to set up a second-round match with former No. 1 Ana Ivanovic.
Another former No. 1, Caroline Wozniacki, had little trouble against 18th-ranked Lucie Safarova, beating her 6-2, 6-2.
Wozniacki, the 2011 Dubai champion, was on course to meet new No. 1 Serena Williams in the quarterfinals.
The Dane, who has returned to the top 10 this year, said she feels her passion for the game has returned.
"For me, I'm just really enjoying the tennis and happy to be out there," Wozniacki said. "I think at one point it was just a job and going and playing, but I feel like I've gotten that hunger back and that enjoyment of the game."
Former U.S. Open Champion Samantha Stosur also eased into the second round, brushing aside 19th-ranked Ekaterina Makarova 6-3, 6-1.
Stosur ran off six games to close out the first set and take a 2-0 lead in the second. She saved three break points to make it 3-1 and a fourth to make it 5-1. It was a sloppy match with the two players combining for 59 unforced errors and only 15 winners, though Stosur converted five of nine break points and won nine of 11 net points.
Williams, who received a first-round bye, became the highest seed at No. 2 when Victoria Azarenka withdrew because of a right foot injury she said she had been carrying for a week, while defending her Doha title and beating Williams in the final at the weekend.
"I tried (to play), but my body says no. I cannot force (it)," Azarenka said. "I cannot put myself in a position to hurt myself. That would just be stupid."
She intended to travel to the United States to prepare for the Indian Wells event next month.
Meanwhile, Williams had her eye on ending her Dubai title drought on her fourth visit. Her big sister Venus has won the tournament twice, but Serena's three previous visits didn't end so well. On her debut in 2005, she had to retire in the semifinals. In 2008, she withdrew before the tournament because of the flu, and in 2009 she fell to Venus in the semis.
This time around, Williams said she was healthy she had a flu bug last week and upbeat even after losing the final in Doha on Sunday.
"I feel so much better this week," Williams said. "I feel like a completely different athlete, a completely different person."
Having become the oldest No. 1 in the WTA's rankings history on Monday, Williams said she was proud to have regained the ranking after a tumultuous few years, and was moving on in search of a 48th career title and more Grand Slam success.
"I feel like I'm at a place where I reached that goal of getting back to the top," she said. "If I stay that would be great and if not then I've done it."
She's set to meet Marion Bartoli in the second round.
Williams showed in 2012 that she is back to her best, winning Olympic gold, Wimbledon, the U.S. Open and the season-ending WTA Championships. That run of success has come as she brought in Frenchman Patrick Mouratoglou whom she credited with bringing "a lot of calm to my game."
"I've obviously had some talent. It's harnessing that talent and moving in the right direction," she said. "The only other (people) able to do that were my parents and Patrick is continuing that job ... It's a good fit. I'm a little intense and I can be a little crazy on court. It's all a balancing act."
Williams said her father remains involved in her career, calling her before every match to offer guidance.