The Gers survived an early scare when hesitation at the back allowed Danny Mullen to collect a stray ball on the edge of the box, but his effort sailed over the bar.
Rangers took the lead in spectacular fashion in the eighth minute as Macleod met Ian Black's cross with an overhead kick that ended up in the top corner.
It should definitely have been 2-0 early in the second half when Richard Foster laid the ball on a platter for Kris Boyd to tuck away inside the box, but the veteran striker scuffed his shot and put it wide.
Livingston remained under intense pressure and they had Jamieson to thank for keeping them in it in the 55th minute, when he produced a superb save to deny Gers captain Lee McCulloch from close range.
The hosts had a couple of chances to sneak an equaliser against the run of play, the best coming in the 68th minute when Michael McKenna tested Gers keeper Steve Simonsen with a header at the back post.
They came close again six minutes later as Kyle Jacobs's header from a tight angle needed a goalline clearance from Black to keep it out, but 1-1 would have flattered Livingston.
The result leaves Rangers in second place in the table, six points behind leaders Hearts, while Livingston are down to eighth.