Murray was romping away at two sets to love before suffering a monumental mid-match slump which saw him lose seven straight games and gift his opponent a route back into the contest.
But after going a break down in the fourth set on a sunny Suzanne Lenglen Court, the 23-year-old finally woke up to win 6-2 6-3 0-6 6-2 in two hours and 33 minutes and set up a last-16 clash with Tomas Berdych.
Murray shrugged off what could have been a more costly blip, saying: "I didn't play a very good set – that was it. That's all that happened.
"I didn't play well, made mistakes. He hit the ball well and that was it. I played a bad set. I'll try to focus on the rest of the match, which I thought was very good."
The world No 4 found himself 4-0 down in the third set when he began to think about the prospect of a fourth.
"I just got off to a bad start and got broken a couple of times," he said. "It's not like you're not trying.
"At double-break down, when you're a couple of sets up, you want to make sure that you have fresh legs for the end of the match."
If that was the plan, it initially backfired as Murray immediately went a break down in the fourth.
But he recovered to win six of the next seven games, a similar feat to that which he pulled off during his second-round victory over Juan Ignacio Chela on Thursday.
"At the beginning of the fourth set, once I got the break back, I started hitting the ball a lot better," he said.
"He's a great shot-maker and very unpredictable, so you need to be focused all the time and I wasn't in the third set."
Murray had lost his two previous meetings with former Australian Open finalist Baghdatis, although they had not played since 2007. The pair had moved in opposite directions following that match, Murray soaring as high as second in the rankings and the Cypriot slumping from eighth to outside the top 150.
But the past year has seen something of a resurgence from the 24-year-old, who is up to 30th in the world and even beat Roger Federer at Indian Wells in March.
After late starts in his two-day second-round win over Chela, quick finishes in the first three matches on Suzanne Lenglen meant Murray was on court yesterday before 3.30pm local time.
That did not prevent a repeat of his early struggles in rounds one and two, the Scot facing a break point and several deuces in his opening two service games.
He held both before being gifted two breaks of his own to clinch the set with ease.
Murray made it six games in a row when Baghdatis dropped serve again in a marathon opening game of the second set, so would have been furious with himself for limply handing the break straight back to his opponent.
But, as in the first set, Baghdatis was unable to close out his service games and Murray broke twice more, the second thanks to a double-fault by the Cypriot.
The second set had lasted a gruelling 56 minutes but even that could not account for Murray's alarming slump in the third.
Baghdatis all too easily secured a triple-break after upping the aggression, Murray seemingly content to concede the set after losing his second service game.
The real concern began when Baghdatis made it four successive breaks of serve and seven successive games at the start of the fourth.
But Murray immediately broke back, snatched the Cypriot's serve again in game six courtesy of a wonderful cross-court backhand before sealing victory with a third break at the first time of asking.
After twilight finishes on days four and five of the tournament, Murray was glad to get off court at a reasonable hour.
Berdych acknowledges he will face a vastly-improved Murray when they square off for the first time in more than four years in the fourth round.
The 15th seed, from the Czech Republic, thrashed John Isner 6-2 6-2 6-1 yesterday and believes both he and Murray have changed so much that their third meeting will seem like their first.
He said: "I have to say that he has changed a lot in the last four years, but I did the same.
"Both of us were pretty young, so we've changed – both of us."
Despite them having not met recently, 15th seed Berdych knows exactly what to expect from Murray.
"He's really strong," he said.
"He can really read the game of his opponent. He's moving well on the court, and he's a really smart player."