Chelsea appeared to come to Switzerland looking for a draw that would clinch its last-16 place, and failed to register a single goal attempt before the Egypt forward's well-earned goal.
Salah sprinted clear of Chelsea's defense to clip a left-foot shot over the advancing Petr Cech and complete home and away wins for the Swiss team against the 2012 European champion.
"I'm very proud. This is the third time I've scored against Chelsea," Salah told Swiss broadcaster RTS. He also scored in last season's Europa League semifinals when Basel lost to the eventual champion.
English opposition again brought the best from Basel, which eliminated Manchester United from the group stage two years ago and beat Chelsea 2-1 in September.
Cech had kept three clean sheets in the competition since that surprising win at Stamford Bridge and looked unbeatable as Basel had six first-half shots on target.
In the third minute Cech crouched to push out a curling free-kick crossed from the left by Taulant Xhaka.
Within a three-minute spell, Cech dived and leaped to tip away 20-meter (-yard) shots by Fabian Frei and Salah, respectively.
Cech's best save denied Salah in the 24th, reaching back with his left hand as he fell to his knees to push the swerving ball up and over the crossbar.
Basel also had two penalty appeals rejected before the break, for an accidental handball by Frank Lampard and left-back Cesar Azpilicueta's timely challenge as Salah shaped to shoot.
Chelsea was livelier in the second half thanks to substitutes Eden Hazard and Fernando Torres, who replaced the injured Samuel Eto'o just before half time, but still failed to create a scoring chance.
Basel lifted the pace in the closing minutes and midfielder Geoffroy Serey Die fired a volley from 25 meters (yards) skidding just beyond Cech's right post.
Defender Fabian Schaer tried his luck from further out in the 85th and forced Cech to dive low to his left to smother the ball.
Schaer then delivered a perfect 60-meter (-yard) diagonal pass into Salah's stride for the decisive goal as the Egyptian left Chelsea right-back Branislav Ivanovic trailing.