The 25-year-old Indian qualifier, ranked 122, had his chances against world number 103 Cypriot, but unforced errors cost him the match 6-7(4), 4-6, 3-6 after two hours and nine minutes on court number eight.
Bhambri is yet to win a round in the main draw of a Grand Slam and the Australian Open is the only Major he has played.
In 2015, when he came through the qualifiers, Bhambri lost to Andy Murray. In 2016, when he made the cut after breaking into the top-100, he drew Tomas Berdych and again failed to go beyond the first round.
It was a good chance for Bhambri to improve his record but a staggering 33 unforced errors to just three of Baghdatis meant that his good work of the past few weeks could not yield him a good result at the big stage.
Bhambri took a break, seconds before the start of the match, much to the amusement of Baghdatis and broke the Cypriot in the very first game, securing the best possible start.
However, the Indian dropped serve in the fourth game with the two players struggling o adapt to the windy conditions.
Bhambri rushed on to the net quite often even as Baghdatis preferred to strike from the baseline. The strategy worked for Bhambri, who yet again broke his rival in the fifth game, only to lose his serve in the very next game.
The Cypriot, a finalist here back in 2006, remained a crowd favourite and the chants of ‘Go-Go Baghdatis’ kept reverberating through the court.
However, it was Bhambri who again took the lead by breaking Baghdatis in the ninth game. Bhambri hit a stunning forehand winner to earn three breakpoints and converted the second when Baghdatis hit a backhand wide.
However, he yet again failed to capitalise and dropped serve while serving for the set. In the tie-break too, Bhambri raced to a 3-1 lead but unforced errors cost him dearly and he went on to lose the opening set.