Top seed Djokovic is chasing a sixth title in Rome ahead of Roland Garros later this month, and needed more than two hours to win a match which featured some magnificent tennis from big-hitting Auger-Aliassime.
Canada's Auger-Aliassime crashed in serves of over 200km/h, hitting 11 aces, and showed a wide range of bold shots in his first ever match against 20-time Grand Slam winner Djokovic.
The 21-year-old, who won his first ATP tournament in Rotterdam in February, even went toe-to-toe in rallies with Djokovic, drawing applause from the Serb early in the match with one particularly deft drop shot.
"He's got the lethal serve, honestly, one thing is to practice another thing is to play against someone in an official match... It was not easy at all for me to return," Djokovic said after the match.
"He's also returning well, he's moving well, he's a very complete player. He did ask me to raise the level."
Djokovic showed touches of class, exemplified by a lob at the end of a thrilling rally which broke Auger-Aliassime's serve in game six of the second set to move the 35-year-old 4-2 ahead.
Auger-Aliassime brought the crowd to their feet by fighting back to take the set to a tie-break, where his brave resistance finally met its end.
Djokovic is top of the bill again on Saturday night for his clash with Casper Ruud where he can claim his 1,000th career victory on way to the final.
Stefanos Tsitsipas made the semis by beating local hero Jannick Sinner 7-6 (7/5) 6-2 in a clash which was stopped for over 10 minutes at match point after a spectator fell ill.
Next up Tsitsipas will face Alexander Zverev, who comfortably saw off Cristian Garin, the only unseeded player left in the tournament, 7-5, 6-2.
It will be the third time in the space of a month that the German will meet the Greek in the last four, with the pair sharing the honours from clashes in Monte Carlo and Madrid.
- Thrilling Jabeur comeback -
Ons Jabeur is in the semis of the women's draw after bouncing back from being a game away from defeat to beat Maria Sakkari 1-6, 7-5, 6-1.
Tunisian Jabeur made history last weekend by winning in Madrid to become the first Arab or African player to claim a WTA 1000 title and let out a roar of delight after a magnificent win at the Foro Italico.
The 27-year-old looked certain to be heading out after being hammered by world number four Sakkari in the first set and going down 5-2 in the second.
Greece's Sakkari was serving for what appeared to be a routine win but from nowhere Jabeur pushed back, breaking serve in game eight before winning the next seven to go three ahead in set three, a lead which she never relinquished.
"In my head at 5-2 I said, You cannot finish those two weeks, those amazing almost three weeks, like this," Jabeur told reporters.
"To back up the performance from Madrid, it's very important for me... It's one of the reasons why I'm here, why I want to continue showing myself on the court."
In the last four Jabeur will face Daria Kasatkina, who went through after Switzerland's Jil Tiechmann retired injured while the Russian was leading 6-4, 3-2.
Earlier Iga Swiatek stretched her winning streak to 26 matches after a straight-sets victory over Bianca Andreescu, 7-6 (7/2), 6-0.
The world number one, bidding for a fifth consecutive tournament triumph, will face Aryna Sabalenka after the third seed won 4-6, 6-3, 6-2 against Amanda Anisimova.