The seventh seed, a semi-finalist at Melbourne Park last year, will meet either China's Peng Shuai or second-seeded Russian Maria Sharapova in the last eight.
Bouchard had looked as if she wanted to spend as little time as possible on court on Sunday as she blasted through the first set in 29 minutes, the opener only lasting that long because Begu held four break opportunities in the fifth game but failed to convert.
Begu, making her first appearance in the fourth round at a grand slam, provided stubborn resistance in the second set, breaking Bouchard four times to force a decider, which the Canadian sealed when the Romanian hit a backhand long.
(Reporting by Greg Stutchbury; Editing by Peter Rutherford)
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