In a match which began at 11.51pm on Saturday night and did not finish until deep into Sunday morning, the former world number one had to draw on every ounce of his trademark fighting spirit to prevail 4-6 7-5 7-5 6-7 (4/7) 6-3.
The match should have finished in the fourth set but Hewitt squandered a match-point, two breaks of serve and a 5-1 advantage, much to the amazement of Baghdatis and the Rod Laver Arena crowd.
Hewitt fell to his knees and collapsed onto the famous centre-court surface after securing the exhausting and emotional contest with a forehand that teased the baseline.
The two-time grand slam winner then warmly embraced his vanquished foe at the net, before soaking up the thunderous reception.
Baghdatis overcame what looked a serious ankle injury early in the third set to mount numerous courageous comebacks.
Out on his feet, the Cypriot wiped away tears as he headed up the tunnel to a standing ovation of his own.
Hewitt described the win - which books him a place in the fourth round against Novak Djokovic - as one of his most memorable and brave.
"In a lot of ways it was not only a physical battle but a mental battle for both of us," Hewitt said.
"Having a match-point and serving for it a couple of times in the fourth set (without success), but being able to bounce back as strong as I did and put that all behind me in the fifth, mentally this will go down as one of my best wins."
Baghdatis was proud of his performance but shattered by the defeat.
"It is tough, but it's a positive thing. I love the game," Baghdatis said.
"It's another time that I went out there, I lost the match, but I came out with so many good emotions."
While Hewitt's triumph all about drama, Federer's 6-7 (5/7) 7-6 (7/1) 5-7 6-1 10-8 win against world number 49 Tipsarevic was pure quality.
And if Rafael Nadal, Djokovic and co thought thought that might hamper his quest for a third straight Australian Open, he had other ideas.
"Honestly for me it's good to play five-setters once in a while. It gives me a lot of information on how I'm feeling, what I still have to work on," said Federer, whose next opponent will be Czech Thomas Berdych, who needed four sets to defeat Juan Monaco of Argentina.
"Just being there in a five-setter is good for me because I'm hardly ever there. It's good for my mindset for the next one.
"My last five-setter was back in Wimbledon I guess, so it doesn't happen every time. When it happens you have to take the most out of it, check it out for yourself and then hope you're ready for the next one."
