The nine-time champion was again in superb form, painting the lines consistently in a 6-1 6-2 6-4 victory over fifth seed Rublev on Rod Laver Arena to set up a clash with unseeded American Tommy Paul.
Djokovic has not been beaten at Melbourne Park since 2018 - part of a 39-match winning run in Australia - and has now matched the sequence of victories of Agassi between 2000 and 2004.
Djokovic said: "I cannot be happier with my tennis. I've been playing very solid. I love playing in these conditions, on this court. Overall I think the scoreline in the first two sets doesn't speak the truth of the match.
"Andrey is a great player. All the important moments and the important shots, I found my best tennis, so that's what I'm most pleased about tonight."
The fifth seed at times did again appear bothered by his left hamstring but it did not affect his ability to hare around the court repelling everything big-hitting Russian Rublev threw at him.
He was more affected by a heckler who called out "Come on Andrey, send him home" during the first set, drawing loud boos from the largely pro-Djokovic crowd and prompting the Serbian to complain to British umpire James Keothavong.
It was Rublev's turn to appeal to Keothavong in the second set when he felt Djokovic should have received a time violation but the Russian, bidding to reach his first slam semi-final at the seventh attempt, was mostly frustrated by the man down the other end of the court.
Djokovic, who has never lost once he has reached the semi-finals at Melbourne Park, was virtually flawless in the big moments, saving all five break points he faced and wrapping up victory in two hours and three minutes.
Earlier, Paul got the better of fellow American Ben Shelton to reach his first grand slam semi-final.
The run of 20-year-old Shelton, in just his second grand slam and on his first trip outside the USA, has been one of the stories of the tournament but Paul used his greater experience to secure a 7-6 (6) 6-3 5-7 6-4 victory.
Speaking before the conclusion of the second quarter-final, the 25-year-old said: "I'm really excited. It wasn't the prettiest match ever today but it was how I needed to play the match. He's a great server. I had to just put in as many returns as I could.
"But I'm really excited, man. It's really cool. I think it's even cooler if I do play [Djokovic]. I probably have a better chance of winning if it's Rublev, but to play Novak here in Australia would be awesome."
Paul, who will break into the top 20 for the first time after the tournament, becomes the first American man to reach the semi-finals here since Andy Roddick in 2009.
With Sebastian Korda also in the last eight, this was the best display by American men at a grand slam for nearly 20 years and appears to herald an exciting era for the country, which has been searching for a successor to Roddick for the last decade.
There is a lot of excitement about Shelton, who has not yet left college, and he might turn out to be the best of the lot, but this was Paul's day.
The American was cheered on by his mother, who hopped on a plane after his fourth-round win and arrived in Melbourne on Wednesday morning.
"It was awesome," he said. "I was actually, when I first heard, pretty nervous because we've had a good thing going here and I didn't really want to change anything.
"After a little bit of thinking, I was like, 'Yeah, I'm in the quarter-finals of a slam. She's sacrificed a ton to get me here. She deserves to be here and deserves to see me win big matches'. I was really excited to see her and have her here."