The 38-year-old Serbian, the highest-ranking player left standing, is getting ever closer to a fifth title in the Chinese financial hub despite struggling physically throughout the week.
Next in the crosshairs of the 24-time Grand Slam champion is world number 204 Vacherot, a qualifier from Monaco who battled back from a set down to stun Holger Rune 2-6, 7-6 (7/4), 6-4.
"Maybe at times (I was) a little too passive but Zizou is a great player, a very talented player, he has some great shots," Djokovic said.
"(Getting through) means the world."
Djokovic broke in the sixth game after an unforced error from the 44th-ranked Bergs.
But just days after he overcame exhaustion, vomiting and an injury scare in his last match against Spain's Jaume Munar, he looked tired and overheated.
He took a puff from an inhaler during a break in play as his team anxiously looked on, but took the first set.
Djokovic broke for a 5-4 lead in the second set, but Bergs broke back after a series of thrilling rallies that drew screams from the crowd.
The umpire had to ask the spectators for quiet multiple times as fans chanted furiously for Djokovic, waving Serbian flags and signs reading "Novak is the GOAT" (greatest of all time).
When Djokovic broke again for a 6-5 lead, the stadium erupted.
Bergs still would not lie down, saving two match points and creating another break point of his own, but Djokovic finally got over the line when the Belgian went long.