Djokovic, who also won his doubles match with Serbian partner Filip Krajinovic, needed only 61-minutes to bypass Stakhovsky and move into the quarterfinals.
"Yes, Orthodox Christmas. It's the biggest holiday that we have," Djokovic said. "Obviously not a holiday for me. I'm working on the court. But it's a big day. So I'm really glad that I was able to perform well and get two wins on this day."
Djokovic, who is playing at the Qatar Open for the first time, won 29 of 34 points when his first serve went in and saved all three break points he faced against Stakhovsky. He next plays seventh-seeded Ivo Karlovic, who beat qualifier Nikoloz Basiashvili of Georgia 7-6 (3), 6-3.
Karlovic became one of three men in tennis history to surpass the 9,000 aces mark earlier this week — he now has 9,041. It's an ability that makes him a tricky challenge for opponents.
"I'm definitely going to try to get some good sleep so my reaction tomorrow is quick, because I'm going to need that, definitely," Djokovic said. "Honestly, I'm still to meet a player who enjoys playing him."
There won't be a singles match between Djokovic and Rafael Nadal at this year's tournament after the Spaniard was upset in the first round.
But the two will face each other in doubles on Thursday, after Nadal and Juan Monaco of Argentina also moved into the doubles semifinals.
"It's going to be interesting to play against each other in doubles, which is not the common case for us, but I look forward to that," said Djokovic.
Third-seeded Tomas Berdych of the Czech Republic brushed aside Slovakian qualifier Blaz Kavcic 6-1, 6-2 to reach the quarterfinals.
Berdych is showing top form this week. In two matches played, he's yet to offer an opponent a break point opportunity.
He now faces sixth-seeded Richard Gasquet, who defeated Simone Bolleli of Italy 6-3, 6-2.
David Ferrer, seeded fourth, won an all-Spanish battle against Fernando Verdasco 6-3, 6-2.
Ferrer, who led Verdasco 5-0 in the second set, was unable to take advantage of one match point on Verdasco's serve in the sixth game, and two match points on his own serve in the seventh game.
He finally won the match on a fourth match point at 30-40 on Verdasco's serve in the eighth game.
Ferrer will play Dustin Brown of Germany, who posted a 7-6 (9), 6-4 win over compatriot Jan-Lennard Struff.
Andre Seppi of Italy and Ivan Dodig of Croatia will compete in the other quarterfinal.
Dodig knocked out Michael Berrer 7-6 (1), 6-2, while Seppi defeated Joao Souza of Brazil 7-5, 6-2.Suggest a correction