Other dangers await after Tomas Berdych and Juan Martin Del Potro also claimed straight set wins, but few in Wednesday's 5,000 capacity crowd would bet against another meeting between the big two, winners of 23 Grand Slam titles between them.
Federer dismissed Spain's Marcel Granollers 6-3 6-4, while Djokovic overcame a late stumble to beat another Spaniard Roberto Bautista Agut 6-1 7-6.
Djokovic was seeking a 15th straight win, his unbeaten streak dating back to October, while Agut has one tour final appearance to his name and the Spaniard was overawed by the Serb's power and movement.
The first set was over in 23 minutes and Djokovic broke for a fourth game in five to go 2-0 up in the second.
The match looked all over, but Agut fought back, breaking Djokovic and saving four match points to take the set to a tiebreak, only for the Spaniard to bow out with a double fault.
"It was not easy to finish the match - he fought very hard and is great competitor," Djokovic said in a courtside interview.
"It was my fault - I made some unforced errors that gave him the chance to come back."
The Serb, 25, now faces Italian Andreas Seppi in Thursday's quarter-finals, while Federer will play Nikolay Davydenko.
The 17-times grand slam champion was adamant he would not take the Russian lightly, despite boasting an 18-2 winning record against the veteran, who has slid down the rankings in recent years.
"I'll never disrespect a guy like Nikolay," said the Swiss. "He's done too much in the game."
After the first five games went with serve, Granollers stumbled, a double fault gifting Federer three break points.
The 31-year-old needed only one, passing Granollers at the net with an arrowed forehand down the line for a 4-2 lead.
Federer, seeking a first title of 2013 and a sixth Dubai crown, was relentless, regularly opting for serve-and-volley tactics, and wrapped up the set with a love game.
"I thought I was sharp, I knew what I wanted to do and I was able to do it," the world number two said in a courtside interview.
"He was going to come in, so I thought I would rather hit a volley than a passing shot - when the courts are so fast, you want to play offensive."
Federer broke again for a decisive 3-2 lead in the second set courtesy of a deep forehand that the Spaniard thought had landed long but his appeal fell flat.
Earlier, world number seven Del Potro pounded Somdev Devvarman 6-4 6-4.
The Argentine saved three break points in his opening two service games but was otherwise largely untroubled in dispatching India's number one.
"It's good, but I need to improve... if I want to have a chance of winning this tournament," Del Potro, 24, said.
Third seed Berdych, 27, is also through to the last eight, beating Germany's Tobias Kamke 7-5 6-1.
The Czech toiled more than the score line would suggest, squandering five set points in the first set against the world number 90 and saving 12 of 14 break points himself.
Rain prevented completion of any second round matches at the Delray Beach International in Florida.
The only singles match to make it on court was eighth seeded Belgian Xavier Malisse against Spanish qualifier Daniel Munoz, but persistent showers forced them off at 2-2.
Second round matches now scheduled for Thursday include top seed American John Isner against Japan's Go Soeda and champion South African Kevin Anderson against Taiwan's Lu Yen-Hsun.
1-Novak Djokovic (Serbia) beat Roberto Bautista (Spain) 6-1 7-6(4)
2-Roger Federer (Switzerland) beat Marcel Granollers (Spain) 6-3 6-4
Nikolay Davydenko (Russia) beat Victor Hanescu (Romania) 6-4 7-6(2)
3-Tomas Berdych (Czech Republic) beat Tobias Kamke (Germany) 7-5 6-1
Daniel Brands (Germany) beat 8-Mikhail Youzhny (Russia) 6-4 6-4
4-Juan Martin Del Potro (Argentina) beat Somdev Devvarman (India) 6-4 6-4
7-Andreas Seppi (Italy) beat Lukas Rosol (Czech Republic) 6-4 7-6(4)
Dmitry Tursunov (Russia) beat Michael Llodra (France) 7-6(5) 2-6 6-3