The Serb prevailed 7-6 0-6 7-6 in a match lasting two hours and 46 minutes and will now go in search of her second title of 2009 after her only previous success came in Marbella in April.
"I definitely got very lucky out there, " a relieved Federer told BBC Sport after winning in three hours and 18 minutes to book a second-round meeting with Serbia's Ilija Bozoljac.
Gasquet won the first two sets but Youzhny finally won 6-7 (9-11) 4-6 7-6 (7-2) 7-6 (7-4) 6-4 in four hours and 52 minutes to earn a second-round clash with unseeded Czech Jan Hazek, who beat American Robby Ginepri in straight sets.
Djokovic won two remarkable battles in Australia, the first against Andy Murray in the semifinals (five sets and nearly five hours) and the second in the final against Rafael Nadal (five sets in five hours and 53 minutes, a record for a Grand Slam final).
WSJ: Novak Djokovic Survives Scare; Belgian Gets Props From Federer
The victory, which took two hours and 29 minutes to complete, was the second in successive tournaments for the Russian over Errani.
The Chinese player was broken to love when first serving for the match but she broke again and converted her second chance with a forehand down the line after two hours and 47 minutes.
The game was delayed 1 hours, 57 minutes at the start, 13 minutes in the second inning and then for another 2:39 before it started again in front of a crowd that began as more than 30, 000 and dwindled to just a few thousand, who waited it out and were rewarded for their patience.
The match, which took nearly four-and-three-quarter hours, included an incredible 71-shot rally in the second set which lasted two minutes and nine seconds.
To put it into perspective, if you were to count one number every second, it would take you 31 years, 251 days, 7 hours, 46 minutes and 40 seconds to reach a billion.
The longest at Wimbledon was the 1989 second round match between Greg Holmes and Terry Witsken in 1989, which took five hours 28 minutes, although arguably the most famous endurance battle was between Pancho Gonzales and Charlie Pasarell in 1969, which took five hours 12 minutes and finished 22-24 1-6 16-14 6-3 11-9 to Gonzales.
Kleybanova, who beat former world No. 4 Jelena Dokic in the first round, led 3-1 in the second set before her 27-year-old opponent rallied to win in two hours and 23 minutes.
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