The 19-year-old, named the WTA's Newcomer of the Year in November, lost a close tiebreaker before dismantling the Australian 6-7 (5/7), 6-2, 6-0 in one hour and 43 minutes.
The hard-hitting Canadian, in her tournament debut, was too hot to handle for Dellacqua, a wildcard in her 11th Australian Open who had never been beyond the fourth round before.
In beating Dellacqua, Bouchard became the first Canadian to reach a Grand Slam quarter-final in 22 years, emphasising her status as one of the most promising players in women's tennis.
The last Canadian to go so far was Patricia Hy-Boulais at the US Open in 1992.
"It was a tough battle but I'm really excited to keep getting better as a tennis player," she said. "I just tried to stay calm and focus on every point."
Dellacqua got the first break in game three when Bouchard misjudged a lob by the Australian, thinking it was going out.
It went with serve until the Canadian pounced in the sixth game to draw level and take the set to a tense tiebreak which the Australian, buoyed by a patriotic crowd at Rod Laver Arena, edged to take a one set lead.
Undeterred, Bouchard bounced straight back with an early break in the second set. She broke again to be 4-1 in front and it went with serve till the match was levelled.
In the third set, the Australian completely lost her game with Bouchard's groundstrokes and dynamic serve proving too much for a player who is better known for doubles.
Australian Open: Eugenie Bouchard races into quarters
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