The Frenchwoman, a semi-finalist on her two most recent visits, misfired badly for six games against the 29-year-old 13th seed on Court One before exploding into life.
With double faults and stray backhands littering her game, the 25-year-old Mauresmo found herself 2-0 and 4-2 down against an opponent who reached the semis at Roland Garros this month.
Likhovtseva then crumbled badly, allowing Mauresmo to take charge. Mauresmo broke back with the aid of a couple of Likhovtseva double faults, then produced a commanding service game laced with exquisite volleys to level at 4-4.
The fifth service break of the first set put Mauresmo ahead for the first time and after serving out the set she romped to victory in 62 minutes.
French Open finalist Mary Pierce made unruffled progress into the quarter-finals, beating Italy's Flavia Pennetta 6-3, 6-1.
The 30-year-old 12th seed broke Pennetta, seeded 25, in the eighth game of the first set and took control on a sun-drenched court three.
Pierce, twice a Grand Slam winner who last reached the quarter-finals at Wimbledon in 1996, swept briskly through the second set, dismantling the Italian's game with her big serve and powerful groundstrokes.
Pennetta, 23, who beat Pierce two years ago in Hyderabad, saved four match-points at 5-1 but the Canadian-born Frenchwoman converted the fifth and can look forward to a showdown with Venus Williams or Jill Craybas in the quarter-finals.
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