Bouchard defeated Begu 6-1 5-7 6-2 on Rod Laver Arena, taking two hours six minutes despite winning nine of the first 10 games.
The cause of supporters group 'Genie Army', Bouchard simply had too many weapons early - noticeably her forehand which produced 23 winners - for Begu.
Begu smashed her previous-best effort at a major - reaching the second round - this week at Melbourne Park, as she strung together three main-draw victories.
Among her notable major results have been beating Caroline Wozniacki at the US Open in 2012, and taking a set off Andrea Petkovic at Wimbledon last year, however there was little evidence early to suggest Sunday's contest would add to that narrative.
Bouchard rocketed to a 5-0 lead, and although she had faced break points in two of her three service games, Begu's resistance seemed fairly futile.
Begu created four break-point chances in the fifth game, and only looked a threat to convert on the last of those but predictably, Bouchard reeled off another forehand winner to deny the world number 42.
Bouchard wrapped up the first set in 29 minutes, with Begu seemingly satisfied to collect her fourth-round losers' cheque worth 175,000 Australian dollars.
Bouchard had not dropped a set coming into the last 16, and when 6-1 3-0 up, even games were at a premium for Begu.
An errant line call that disadvantaged the Romanian seemed to spark her in the fourth game of the second set, though, and she would go on to break the Bouchard serve for the first time in the match to reduce her deficit to 1-3.
It turned out to be a match-changing break, as Bouchard's serve - and overall game - hit the skids.
A pile of Bouchard unforced errors - 10 in four games - helped Begu gain a foothold, and the 24-year-old would take a 4-3 lead.
Begu's slight resistance turned full match challenge when Bouchard's forehand went long yet again to hand Begu the chance to the serve for the set.
Bouchard broke back to love, as she regathered herself and discovered her range once more, but a net cord caused a double fault on set point to see Begu square the match.
The Wimbledon finalist was having few dramas on return against Begu - it was her service games that were letting her down, but the fourth game of the third set was a pressure reliever.
Bouchard had only held to love or 15 twice prior, so when she went 40-0 up and eventually held to 15, it was more telling than a regulation service game.
She only dropped two points on serve thereafter, as Bouchard survived her first scare of the tournament.