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, 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 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," he said.
"I think from the first minute I was getting signs that my team was tired (and) the goal is ridiculous."
His frustration contrasted with the delight of opposite number Murat Yakin, who described Basel's victory as "wonderful".
"At half-time I congratulated my team for playing fantastic football," Yakin said.
"It was a more even game after the restart, but the team knew it was possible and we kept fighting to win the three points," he said.
Mourinho had made just one change to his side, with Willian replacing Eden Hazard, who dropped to the bench alongside the fit-again Fernando Torres.
And his 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 outstanding 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.