Kai Havertz’s stunning strike 26 minutes from time proved the difference between the sides after Junior Adamu cancelled out Mateo Kovacic’s opener for the Blues.
Victory extended Chelsea’s unbeaten run under Potter to nine games.
The former Brighton boss was appointed at Stamford Bridge after Thomas Tuchel’s final game in charge ended in a 1-0 defeat to Dinamo Zagreb to open Group E.
A 1-1 draw at home to the Austrian champions in Potter’s first game in charge then left Chelsea with a mountain to climb.
But back-to-back victories over AC Milan put their European campaign back on track and another win opens up an unassailable four-point lead over Salzburg.
Anything other than a victory for Dinamo Zagreb over AC Milan later on Tuesday will also guarantee top spot in the group for the 2021 European champions.
“That was a fantastic performance from the players,” said Potter.
“I’m delighted to go through with a game to spare considering where we started the competition.”
Chelsea’s bright start under Potter has been undermined by failing to turn their approach play into goals and that was the case once more in a dominant first 45 minutes.
Salzburg had not suffered defeat in their first four Champions League games of the season, but were grateful to goalkeeper Philipp Kohn for keeping the score down before the break.
Kovacic’s fine strike into the top corner gave Kohn no chance as he opened the scoring on 23 minutes.
But the German stopper won a personal duel with Pierre-Emerick Aubameyang with three stunning saves to deny the former Arsenal captain.
– Havertz’s quality counts –
Havertz also had a header from close range parried by Kohn as Chelsea clicked into an attacking rhythm rarely seen during Potter’s first two months in charge.
The English side looked like they may live to regret not taking those chances as Salzburg equalised four minutes into the second half when Adamu converted Maximilian Wober’s teasing cross in behind the Chelsea defence.
However, the visitors immediately responded as Kohn rushed out to deny Aubameyang once more before Adamu cleared Jorginho’s header off the line from a corner.
Chelsea finally got the second goal their performance deserved when Havertz cushioned Christian Pulisic’s pass into his path before curling in off the underside of the crossbar.
“It makes my job a lot easier when they can do that,” added Potter on just Havertz’s third goal of the season.
“He has been fighting and trying his best, for him to score that and help us through is fantastic.
“His quality is there, he’s an incredibly talented player. He’s been nearly there. Scoring a goal like that is bound to help.
“I wouldn’t know, I couldn’t score a goal like that, but it’s a fantastic strike and an indication of the quality he has.”
But Potter’s men had to withstand a late onslaught from Salzburg to hold out for all three points.
The in-form Kepa Arrizabalaga was forced to fly low to his left to turn Benjamin Sesko’s effort behind.
Kepa is finally beginning to deliver on his price tag as the world’s most expensive goalkeeper following his move from Athletic Bilbao in 2018.
But the Spaniard was bailed out by Thiago Silva’s goal line clearance after he was beaten to a cross by Strahinja Pavlovic.
AFP