In challenging conditions, Djokovic was made to work harder than the scoreline suggests by his Belgian opponent, who actually registered more winners than the four-times champion but was undone by unforced errors.
"First encounter with Bergs, a great guy. Obviously a lot of firepower in his game. He played a good game," Djokovic said after his win on Thursday.
"Again, I was a little bit too passive. Just very challenging conditions these days for all the players. Just trying to stay alive on the court and glad to overcome this hurdle."
In the opening set, Bergs showed remarkable resilience by saving five set points to frustrate Djokovic, before the Serbian finally closed it out with an unreturned serve.
The second set proved even more gruelling, with some breathtaking rallies that left Djokovic hunched over his racquet on several occasions.
The highlight came during one extraordinary rally where Djokovic gave Bergs five golden opportunities to seal the point with overhead shots and volleys at the net, only to somehow emerge victorious and bring the crowd to their feet.
Djokovic eventually sealed victory on his third match point, advancing to face Vacherot in what promises to be an intriguing semi-final.
"Amazing story for him. This tournament has taken out top players of the world... it's really impressive what he's doing," Djokovic said.
Vacherot pulled off another huge upset, stunning 10th seed Holger Rune 2-6 7-6(4) 6-4 to become the second-lowest ranked player ever to reach an ATP Masters semi-final.
The Monegasque qualifier's remarkable run ensures he will crack the top 100 next week after spending half of last year sidelined with injury. Even playing in Shanghai was a question mark when he arrived.
"I didn't even come as a qualifier, I came as an alternate. I wasn't sure to even play the qualifier," Vacherot said.
"Coming back from six months out is always not easy. You cannot just come back and win tournaments right and left. You've just got to fight your way through a little bit."
Vacherot's stunning journey in Shanghai, which now includes wins over four seeds, looked doomed early on as Rune raced through a physically demanding opening set while the underdog felt his "lungs were screaming".
But the tide turned dramatically in the second-set tiebreak when Vacherot produced a stunning backhand crosscourt winner to take a 5-4 lead.
Rune crashed his next shot into the net and could only watch as Vacherot fired a forehand winner down the line on set point to level the match.
Rune struggled physically in the decider, requiring the trainer twice to massage his leg. Vacherot then capitalised to complete a memorable victory in a contest lasting a minute shy of three hours.
"It would mean a lot to play at least one of the guys of the 'Big Three' in my career," Vacherot said on playing Djokovic.
Vacherot's cousin Arthur Rinderknech is also in the quarter-finals, with the Frenchman set to play Felix Auger-Aliassime on Friday, while Australian hope Alex de Minaur meets No.16 seed Daniil Medvedev.
