Mirza had taken several months off from tennis to focus on her personal life before controversially marrying Malik in April.
The tense nature of relations between India and Pakistan meant political groups in her homeland protested against the 23-year-old's decision to marry Malik.
Mirza's picture was burnt on the streets of Bhopal, where activists from the right-wing Hindu nationalist Vishwa Hindu Parishad party vowed to stop her competing in the Commonwealth Games in Delhi in October.
But, with the storm of controversy dying down, Mirza has been able to return to tennis and she quickly rediscovered her rhythm on the Birmingham grass-courts as she swatted aside Chan, a 20-year-old from Taipei.
Britain's Laura Robson, who last week had to offer an abject apology for an interview she had given where she labelled some of her fellow players as 'sluts', advanced to the second round when her Swiss opponent Stefanie Voegele retired in the second game of the second set.
Robson, a junior Wimbledon champion in 2008, had taken the first set 6-4.
Lucic walked away from tennis in 2003 after falling out with her father and the 28-year-old, who had been ranked as high as 32nd in the world, is only now beginning to rebuild her career.
There was also success for Michelle Larcher De Brito, the Portuguese teenager best known for the ear-piercing scream she lets out before most strokes.
Larcher De Brito moved into the second round with a 6-2, 4-6, 6-3 win over Ekaterina Dzehalevich.
Source: AFP Global Edition