Salah latched on to a long ball forward by Fabian Schaer and outpaced Branislav Ivanovic as he cut in from the left before finishing past Petr Cech to hand the hosts a thoroughly deserved victory and keep their hopes of joining Chelsea in the last 16 alive heading into the final round of matches.
It was Basel's second win against the Premier League club in this season's competition and propelled them into second place in the group above Schalke 04, who only managed a goalless draw away to Steaua Bucharest in the night's other match.
That result was good news for Chelsea too, as they are now certain to take one of the top two spots in the group.
However, they still have work to do if they are to clinch first place, with a home game against Steaua still to come, and manager Jose Mourinho was deeply disappointed with what was a flat display by his side, three days after they had thumped West Ham United 3-0 away in the Premier League.
"The only positive thing is that we go through, but we go through because Schalke did not get a result," said the Portuguese.
"But I do not want to take any credit from the opponent as they were better than us. "I think from the first minute I was getting signs that my team was tired...The goal is ridiculous."
Mourinho had made just one change to his side, with Willian taking the place of Eden Hazard, who dropped to the bench alongside the fit-again Fernando Torres. That meant that Cesar Azpilicueta again started at left-back with Ashley Cole also only a substitute.
And Mourinho's decision not to rotate his squad meant they also lacked freshness, as the Blues produced a lacklustre performance in a game played in sub-zero temperatures.
That was reflected in the statistics, with Basel having more of the possession and all the chances in what was an oustanding first-half showing in particular.
Chelsea only came close to scoring once in the opening period, when Samuel Eto'o, diving in at the back post, just failed to connect with an Ivanovic cross from the right, while Basel brought the best out of Cech on several occasions.
The Czech goalkeeper needed to be alert to keep out Marco Streller's header from a Valentin Stocker free-kick inside three minutes, and that early scare for Chelsea set the tone for the opening half an hour.
Basel, with seven players aged 24 or under in their starting line-up, were by far the livelier of the two teams, and Cech was also called into action to turn Fabian Frei's shot from long range around the post before twice denying the talented Egyptian Salah.
In between, Basel were also denied a penalty by French referee Stephane Lannoy, who let play go on when the ball came off Frank Lampard's arm inside the area.
Chelsea's first half was summed up when Eto'o was stretchered off following an innocuous-looking challenge with Serey Die during a rare foray forward, and Torres took his place.
There was a feeling that Chelsea had been let off the hook as they went in at the interval on level terms, and their defence was rarely troubled after the restart, with the exception of a Serey Die volley that flashed just wide after a poor clearance by John Terry.
They also looked brighter going forward once Hazard had come on in place of the ineffective Oscar but, remarkably, Basel 'keeper Yann Sommer did not have a save of note to make all evening.
And Basel finally got the breakthrough they merited with three minutes left when Salah - who scored in their 2-1 win at Stamford Bridge in September - sent the St Jakob-Park crowd into raptures.