Roger Federer defied age and logic once again to save seven match points before beating unseeded American Tennys Sandgren in a sensational Australian Open quarter-final.

The out-of-sorts Swiss fought off form and fitness concerns before winning 6-3 2-6 2-6 7-6 (10-8) 6-3.

World number 100 Sandgren looked set to be the lowest ranked player to reach the Melbourne semi-finals since 1991.

But third seed Federer battled to win in three hours and 28 minutes.

The 20-time Grand Slam champion, who had a medical timeout in the third set for a groin problem, will play Serbia's Novak Djokovic in the last four after the defending champion beat Canadian 32nd seed Milos Raonic.

Federer calmly raised his racquet in the air, taking the acclaim of a buzzing crowd on Rod Laver Arena, after Sandgren dumped a forehand into the net on the Swiss' first match point.

The fatigued pair shook hands at the net, Sandgren wishing his rival luck before trudging off to a standing ovation from the 15,000 fans barely able to believe what they had seen.

"You've got to get lucky sometimes I tell you that," Federer said. "Sometimes you're not under control.

"I just hoped he wouldn't smash a winner, just keep the ball in play and if he does miss one of two then see what is going through his mind.

"As the match went on I felt better and the pressure went away.

"I didn't deserve this one but I'm standing here and I'm obviously very happy."

Federer hangs tough again

With 38-year-old Federer well into the twilight of his career, he has to carefully manage his workload to preserve himself for the things which matter most: the Grand Slams.

And that appears to have worked in his favour as he came through another gruelling match at Melbourne Park after being on the brink of defeat.

Federer had not played since the ATP Finals in November, save for a few exhibitions in Central and South America, and has been helped by a favourable draw at Melbourne Park.

All four of his matches have been against unseeded players, although they did not all prove to be as straightforward as he would have hoped.

Only a fightback from 8-4 in the 10-point match tie-break against Australian John Millman stopped him falling in the third round.

In the next round he was pushed by Hungary's Marton Fucsovics before winning in four.

Again he was a shadow of himself for most of the contest against Sandgren before rediscovering his best form when it mattered most.

After a fairly routine opening set, Federer's level dipped considerably as his unforced error count more than doubled in the second and mistakes continued to fly off his racquet in the third where he had treatment for a groin injury when trailing 3-0.

His serve speeds slowed considerably and he walked gingerly to his chair at the changeovers.

The six-time champion - and the pro-Federer crowd - became increasingly edgy. Sandgren grew in confidence.

But once Federer overcame his injury, he improved considerably in the fourth and dominated in the fifth to earn another memorable victory.

"My groin and leg started to tighten and I struggled in defence," said Federer who has never retired during a career which has seen him play 1,512 matches.

"I don't like to call the trainer because it is a sign of weakness. The best thing with the groin is you go off court and people don't know what it is.

"I believe in miracles, there could be rain, there could be other stuff. It wasn't bad enough where thought it would get worse, just sore.

Sandgren made to pay for missed chances

While Federer was off-colour for long spells, Sandgren looked to capitalise with big serving and booming groundstrokes.

The American thumped down 27 aces and cracked 73 winners, but it was still not enough to reach his first Grand Slam semi-final.

The magnitude of what he was on the verge of achieving suddenly dawned on him when his first three match points arrived at 5-4 in the fourth set on Federer's serve.

All three disappeared as a result of unforced errors from the American's racquet, with Federer keeping himself in the rallies, putting balls back into the court to see if his opponent would crack.

Another four match points came Sandgren's way in the tie-break, but two more unforced errors ended energy-sapping rallies, either side of a missed backhand and then a wonderful volley from Federer.

From that point Sandgren was, seemingly, overcome by mental doubt and physical strain. He never recovered.

"I thought l would let him finish me off in style and he didn't do that. I was incredibly lucky," said Federer, who is the oldest man to reach the Australian Open semi-finals since Ken Rosewall in 1977.

A rare Federer outburst & Sandgren clattered by ball girl - the match that had everything

A sign of how flustered Federer had become came in the third game of the third set when the Swiss was given a rare code violation for an audible obscenity.

He had just failed to take a third break point at 2-0 down.

The line judge behind the baseline reported Federer, who was outraged by the warning as he lost the game.

"What did I say," he asked chair umpire Marijana Veljovic. She replied: "I can't repeat it."

That was not the only unusual incident in a match packed full of entertainment.

Later on Sandgren - whose knee has been strapped for his matches here - was accidentally clattered by a ball girl running past him at a changeover in the fourth.

"That was not a distraction. That was physically painful," said a sombre Sandgren, who recovered from the incident to win the next three points.

He was, however, distracted by somebody commentating behind him when he served and a crackling public address system, and a phone rang twice while Federer prepared to deliver from the baseline.

It truly was a match that had everything.