The British number one, who along with the Cypriot was knocked out of the doubles competition on Thursday, was quick out of the blocks and looked full of confidence as he raced into a 4-0 lead which he never looked in danger of relinquishing.
Baghdatis proved a trickier proposition at the start of the second set but when Murray clinched the first break in game five it was only a matter of time.
The world number four had slow starts in his opening two rounds, losing the first sets against both Mikhail Kukushkin and Gilles Muller, but looked sharp from the outset on Pat Rafter Arena today.
His powerful forehands troubled the 2006 Australian Open runner-up from the word go, a Baghdatis double fault handing Murray the opening game before the Scot clinched a second break to love in game three.
Baghdatis got on the board by holding serve at the third time of asking, but never looked like threatening the Murray serve as the Briton held firm to win the opening set.
The start of the second set was a much tighter affair than the first, Baghdatis holding his opening two games and forcing a break point in game four which Murray saved thanks to his serve-and-volley tactics.
Murray made his rival pay for that missed opportunity by securing a break of his own in the next game, and consolidated that advantage by holding serve - saving another break point - to go 4-2 up.
Another break followed as crowd favourite Baghdatis ran out of steam, Murray completing an impressive win with minimum fuss.
Next up for the Scot will be a semi-final meeting with Uzbekistan's Denis Istomin or Australian eighth seed Bernard Tomic, who go head to head in the final quarter-final.
Second and third seeds Gilles Simon and Alexandr Dolgopolov claimed their places in the last four on Thursday.