World number four Murray was in the comfort zone after winning the opening two sets before completely losing the plot, gifting his opponent seven successive games and a route back into the match.
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 to set up a last-16 clash with Tomas Berdych or John Isner.
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 to as high as second in the rankings and the Cypriot slumping from eighth to outside to top 150.
But the past year has seen mini-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 Juan Ignacio Chela, quick finishes in the first three matches on Suzanne Lenglen meant Murray was on court today 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 and was gifted a break of his own in game five thanks to two bad errors from the Cypriot.
Murray's tactics were similar to those against Chela, the 23-year-old happy to trade groundstrokes with his opponent from the back of the court and wait for mistakes.
Baghdatis obliged again in game seven to hand him another two break points, Murray taking the second.
Serving out the set proved little more than a formality for the Scot, who claimed a fifth straight game.
Baghdatis dropped serve again in a marathon opening game of the second set, Murray finally breaking at the fifth time of asking when the Cypriot netted.
He would have been furious with himself, therefore, for limply handing the break back to his opponent in the very next game.
Murray fought back from 30-0 down to fashion two break points in game three but could not convert.
Baghdatis was looking disinterested one minute and all too eager the next, shipping and saving another two break points in an interminable fifth game before Murray punished a third with a fine forehand winner.
This time, he held his own serve and again before a delightful backhand volley set up two set points on the serve of Baghdatis, who double-faulted for a sixth time to gift him a 2-0 lead.
The second set had lasted a gruelling 56 minutes and Murray endured an alarming slump at the start of the third, falling 0-40 down before netting to hand Baghdatis a break to 15.
The Cypriot crashed himself from 30-0 up to 30-40 in the next before holding, but Murray was now really struggling on serve, Baghdatis all too easily securing a double-break and a 3-0 lead.
The 25th seed had upped the aggression and Murray was failing to respond, surrendering serve again for a 5-0 deficit.
Having seemingly given up on the set, Murray allowed his opponent to serve it out and regrouped for the fourth.
If that was the plan, it immediately backfired as he was broken courtesy of a double-fault.
Baghdatis showed some charity in the next to allow him to end his seven-game losing streak by breaking straight back.
Despite being taken to deuce, Murray then held for the first time in five service games.
Indeed, both men found holding uncharacteristically straightforward until Murray snatched Baghdatis' serve in game six courtesy of a wonderful crosscourt backhand the Cypriot simply could not handle.
Baghdatis was not even able to force his opponent to serve for the match, a Murray drop shot clinching it at the first time of asking.
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Baghdatis dropped serve again in a marathon opening game of the second set, Murray finally breaking at the fifth time of asking when the Cypriot netted.
He would have been furious with himself, therefore, for limply handing the break back to his opponent in the very next game.
Murray fought back from 30-0 down to fashion two break points in game three but could not convert.
Baghdatis was looking disinterested one minute and all too eager the next, shipping and saving another two break points in an interminable fifth game before Murray punished a third with a fine forehand winner.
This time, he held his own serve and did so again before a delightful backhand volley set up two set points on the serve of Baghdatis, who gifted him the set 6-3 and a two sets to love lead.
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The second set had lasted a gruelling 56 minutes and Murray endured an alarming slump at the start of the third, falling 0-40 down before netting to hand Baghdatis an early break.
The Cypriot faced a break point in the next before holding, but Murray was now really struggling on serve, Baghdatis easily securing a double break and 3-0 lead.
The 25th seed had upped the aggression and Murray was failing to respond, surrendering serve again for a 5-0 deficit.
Having seemingly given up on the set, Murray allowed his opponent to serve it out and regrouped for the fourth.
If that was the plan, it immediately backfired as he lost his serve courtesy of a double-fault.
Baghdatis showed some charity in the next to allow him to end his seven-game losing streak by breaking straight back.
Despite being taken to deuce, Murray then held for the first time in five service games.
Indeed, both me found holding uncharacteristically straightforward until Murray snatched Baghdatis' serve in game six courtesy of a wonderful crosscourt backhand the Cypriot simply could not handle.
Baghdatis was not even able to force his opponent to serve for the match, a Murray drop shot clinching it at the first time of asking and sealing a 6-2 6-3 0-6 6-2 win.
