World number one Rafael Nadal also won through to the last eight where he will face rising Bulgarian star Grigor Dimitrov in Wednesday’s quarters in the top half of the men’s draw.
Record Grand Slam champion Federer had few problems with 10th-seeded Frenchman Tsonga, winning 6-3, 7-5, 6-4 in one hour 52 minutes in the fourth-round evening match at the Rod Laver Arena.
Federer’s comfortable victory over the 2008 Melbourne finalist set up a rematch of last year’s semifinal which Murray won in five sets.
Wimbledon champion Murray was taken to four sets and smashed his racquet in frustration before finally ending ‘lucky loser’ Stephane Robert’s dream run.
In Monday’s other men’s action, Nadal received two time violation warnings for slow play as he fought off a spirited challenge from Japan’s Kei Nishikori, 7-6 (7/3), 7-5, 7-6 (7/3) in three hours 17 minutes.
Nishikori, showing the effects of working with new coach Michael Chang, pressed Nadal all the way and had his chances to take sets off the top seed only to lose the big points in the match.
Dimitrov became the first Bulgarian man to qualify for a Grand Slam quarter-final and said he fancied his chances against Nadal after ousting unseeded Spaniard Roberto Bautista Agut 6-3, 3-6, 6-2, 6-4.
Federer has now reached his 11th consecutive quarter-final at the Australian Open and equalled the record of 41 straight Slam quarters held by American Jimmy Connors.
Federer’s quarter-final with Tsonga went to five sets in last year’s Australian Open but the Swiss sixth seed was always in command with his aggressive display, dominating the net exchanges to win in straight sets.
“I’m very pleased," Federer said.
"I didn’t get broken and that was against a great player," he said.
"So, yeah, I’m extremely happy how things went for me tonight,” Federer said.
“I was able to play my game, offensive, mix it up, come to the net," he said.
"I was surprised that things worked out for me.”
Murray easily won the first two sets but passed up three match points in the third — and totalled a racquet in a fit of pique — before beating Robert, 6-1, 6-2, 6-7 (6/8), 6-2.
Robert, at 119, has a ranking 115 places below the Scot, a three-time finalist in Melbourne.
The Frenchman, 33, was the first lucky loser — someone who loses in qualifying, but gains a place via a withdrawal — to reach the tournament’s last 16, and the oldest man since Andre Agassi in 2005.
“He’s a fun player to watch but not fun to play against,” said Murray. “He made it very tricky for me, he plays all the shots very unorthodox.”
Nadal called for more understanding from umpires over time warnings as he praised Nishikori’s performance.
“Kei played a fantastic match," Nadal said.
"Just a few mistakes in some moments that were tough ones," he said.
"But he played very aggressive, he went for his shots.” Nadal said.
“He came on court with determination to take the ball very early and go for the winners," he said.
"It was just a tough match, a very important win.”
Nadal was angered when he was called for an unsettling second time violation in between points by chair umpire Eva Asderaki at deuce while 4-4 at a critical time in the third set, but he regained his composure to go through on his second tiebreaker.
The 13-time Grand Slam champion did not dispute the 20-second rule in between points at Grand Slams, but rather took issue with the timing of the warning.