A sparkling rally on set point finally ended with Wawrinka hitting a forehand wide.
"It's very important to me," said Djokovic, who dropped to his knees and thumped his chest in celebration after winning the first set.
"I'm a very emotional player on the court and very emotional person off the court as well. As you could see by the reaction, it was probably the most important point of the game.
"We had such long rallies. But in the end I won, and that's what matters."
Rafael Nadal |
"You know, 6-5, he was serving for the set, it could go either way.
"For us it was really difficult, because we had such long rallies and points. It is not easy to hold on, but obviously I was a bit lucky in the first set."
Despite his inconsistencies, especially in the opening set, Djokovic was pleased to come away with the win.
"This was really a test for me," the Serb said.
"You know, we played over two hours only two sets. We had such long rallies. But in the end I won, and that's what matters."
Nadal continues title defence
Djokovic will now face defending champion Rafael Nadal in the semis after the Spaniard overcame American James Blake 7-5, 3-6, 6-3 in their quarter-final.
Nadal, the second seed, earned his first victory over Blake in four career meetings after being beaten by the American in all three of their previous encounters, the last at the Tennis Masters Cup in 2006, when he dropped a second-set tiebreaker 7-0.
This time, the world number two earned the only break of the third set in the eighth game to take 5-3 lead, and served out the match to love.
Nadal fired down his sixth ace of the contest on match point and dropped to his knees in celebration.
The Spaniard beat Djokovic in the Indian Wells final last year.
The men's quarters continue on Friday when Roger Federer, top seed and world number one, takes on Germany's Tommy Haas, while seventh-seeded Argentinian David Nalbandian plays unseeded American Mardy Fish.