The top-ranked Serb needed just 55 minutes to defeat Kubot, a 64th-ranked qualifier.
"It was really a comfortable match for me," said Djokovic. "It was kind of expected at the start of the tournament that I would struggle with my rhythm ... but today I have no complaints with my game."
He served eight aces and dropped just 10 points in eight service games.
"I just started serving two or three days before my first match (on Tuesday). I can still feel on the second serve that I'm struggling a little bit," Djokovic said.
Djokovic improved to 66-3 this year in his first tournament since winning the U.S. Open in September.
Djokovic will play Marcos Baghdatis of Cyprus in the quarterfinals on Friday.
Roddick next plays Roger Federer, a four-time champion at his hometown event, in a meeting of the remaining seeded players to reach the last eight.
"I think the only other tournament we have played so many times is Wimbledon," said the American, who is 0-3 against Federer in Basel.
Mayer beat Ivan Ljubicic 6-3, 6-1 for his first victory in four matches against the big-serving Croat.
Mayer is enjoying a great year at age 28. He won his first tour title in September, at Bucharest, Romania, and beat Rafael Nadal at the Shanghai Masters last month.
"It gave me a lot of confidence, that's for sure. It's maybe a great chance to take (my ranking) into the top 20," Mayer said.
Wawrinka will face Mayer on Friday, after he also rebounded from a one-set deficit to beat Robin Haase of the Netherlands 4-6, 6-1, 6-2.
Nishikori continued to make good use of his wild-card entry, beating Andreas Seppi of Italy, 6-3, 7-6 (4).
The 32nd-ranked Japanese had beaten fourth-seeded Czech Tomas Berdych in the first round.
Kukushkin reversed his qualifying-round loss against James Blake to beat the American 6-3, 6-4. He got into the main draw after Sunday's defeat as a lucky loser and will play Nishikori in the quarters.