After the top seed's "match to forget", Federer looked in a hurry to brush off David Goffin when they came on court just before 11:00 pm -- and he duly thrashed the Belgian 6-2, 6-1, 6-4 in 88 minutes.
There was a big contrast between defending champion Djokovic, who is going for a record-equalling sixth Australian Open title, and the 34-year-old Federer, who hasn't won in Melbourne since 2010.
While Simon patiently pushed and prodded Djokovic during his 6-3, 6-7 (1/7), 6-4, 4-6, 6-3 win, Goffin couldn't lay a glove on Federer as the clockwork Swiss purred through.
Berdych also had to come through a marathon five-setter as he outlasted Spain's Roberto Bautista Agut 4-6, 6-4, 6-3, 1-6, 6-3 to reach the last eight for the sixth year in a row.
More From This Section
By comparison, Nishikori's win was a stroll as he crushed Jo-Wilfried Tsonga in straight sets, and he may fancy his chances against Djokovic after the Serb's flustered performance against Simon.
"Actually, it gives me great joy to know that I can't get worse than that, than what I played today," Djokovic said.
He was at a loss to explain his tactic of continually trying drop shots, which either found the net or were returned with interest by the fast-moving Simon.
"Sometimes you have a brain freeze, if I can call it that way," the 10-time Grand Slam champion said.
- Frustrations show -
=====================
Tuesday's clash will be Djokovic and Nishikori's first Grand Slam meeting since the 2014 US Open semis, when the Japanese player won in four sets to become the first Asian man to reach a major singles final.
The Serb took nearly an hour to win the first set and his frustrations were palpable as Simon took the second on a tiebreak, ending an unbeaten sequence of 26 sets stretching back to the World Tour Finals.
He got the crucial break to take the third set but the persistent Simon took it into a fifth after he broke late in the fourth set, amid another flurry of errors from the Serb.
Djokovic got two service breaks in the fifth set only to be broken back once, but he finally clinched victory on his third match point with a backhand winner.
"Today was one of the best matches I've had this week. Played good tennis, through in three sets," beamed Nishikori, who showed no sign of his troublesome right wrist injury.
Berdych was also taken the distance by a dogged Bautista Agut before he finally put him away after three hours, 18 minutes to reach the last eight.
But Federer was in no mood to hang around in the evening's final match, watched by Australian great Rod Laver in the arena that bears his name.