The Ibrox side were looking to record their first win on the road since February’s 4-1 triumph at St Johnstone, but Gary Holt’s team inflicted just Gerrard’s second defeat in 18 games as manager to leave his side eight points adrift of runaway Ladbrokes Premiership leaders Hearts.
Menga struck seven minutes before half-time and a Rangers line-up short on inspiration never looked like levelling.
While Rangers now drop down to sixth spot, Livingston climb to third and only behind Hibernian on goal difference. And there could be further trouble in store for Rangers after linesman Calum Spence had to be treated for a head wound after an object was apparently thrown from the away end.
Gerrard made eight changes to the side which swept past Ayr into the Betfred Cup semi-finals. Holt, on the other hand, went with the same Livi 11 which took a point off Hearts last week and he would have been delighted with the way his well-drilled outfit suffocated their opponents at every turn.
Rangers may have dominated possession but Livingston did more with the ball they had. They should have scored just two minutes in when Menga’s first-time lay-off sent Scott Pittman through on Allan McGregor’s goal but the midfielder tugged his big chance wide.
A deep Keaghan Jacobs free-kick then found skipper Craig Halkett but his defensive partner Alan Lithgow could only stab wide at the back post. Rangers did grab control but there looked to be more hope than belief about their play in the final third.
Alfredo Morelos certainly felt he should have had a penalty on the quarter-hour mark as he and Steven Lawless collided but referee Steven Thomson waved away the Colombian’s appeals.
Rangers managed just two first-half shots on target but former Ibrox goalkeeper Liam Kelly was barely troubled by Lassana Coulibaly’s header and a tame Ovie Ejaria strike.
While Livi’s 34th-minute opener may have been against the run of play, no one can deny they deserved it for their graft alone. Again Rangers failed to deal with a deep Jacobs set-piece. Scott Arfield did not react as Declan Gallagher slotted the ball to Scott Robinson, who kept his cool to pick out Menga for a simple finish.
Gerrard acted at the break as he swapped Coulibaly for Ryan Kent and the switch almost provided an immediate impact as the winger slotted in Morelos but the impressive Kelly raced from his line to block.
But Livingston continued to suck the visitors into playing exactly the type of game they enjoy, with Lithgow and Halkett sweeping up long ball after long ball. Holt’s side could have stretched their lead in the final stages but McGregor stood up well to deny Jacobs before Lithgow headed wide from the corner.
Rangers substitute Glenn Middleton had another late claim for a penalty as he tangled with Jacobs but again Thomson looked away.