Captain James Tavernier converted his sixth penalty of the season early in the match after Ross Millen’s handball.
Connor Goldson, Joe Aribo, Ryan Kent and Cedric Itten had chances to extend the visitors’ lead but Steven Gerrard’s side had to settle for a solitary goal.
For Kilmarnock, it was a second defeat in two games.
Chris Burke and Eamonn Brophy tested Allan McGregor from free-kicks but Alex Dyer’s side could not force a leveller and end the weekend sixth.
Gerrard’s side now have a significant cushion over title rivals Celtic, albeit having played two games more than Neil Lennon’s side.
The league leaders started on the front foot and Joe Aribo came close with a left-foot strike that rose just over.
It was a Borna Barisic free-kick that led to the penalty, with his delivery coming off Goldson’s head, then Millen’s outstretched right hand.
It gave referee Andrew Dallas little option but to point to the spot and Tavernier swept the ball to Danny Rogers’ right for his 10th goal of the season.
Goldson headed a Tavernier corner wide and Aribo had another effort deflected away as Rangers dominated the remainder of the first half.
Kilmarnock’s only attempt was a long-range free-kick that Burke fired into the arms of former Ibrox team-mate McGregor.
Alan Power tried to change the course of the match early in the second half with a skilful juggle and volley after getting beyond his team-mates, but McGregor collected.
Kent tested Rogers and the goalkeeper also denied Itten from close range after substitute Steven Davis sent in a teasing cross.
In between those chances, Rory McKenzie pulled a volley wide and a late free-kick afforded substitute Brophy a shooting chance but McGregor beat clear.
Man of the match – Connor Goldson
What did we learn?
Lech Poznan had limited Rangers to very few chances in the Polish side’s midweek Europa League defeat at Ibrox and Kilmarnock’s approach was similar, packing central areas every time the visitors came forward.
However, in the first half, that invited Rangers to make full use of the flanks and limited the hosts’ numbers when they went forward.
Rangers played with intensity before and after the opening goal but could not put real distance between themselves and Kilmarnock.
Recent memories of recovering from a goal down to beat Rangers 2-1 gave Dyer’s side hope the longer the score stayed at 1-0, but history would not repeat itself on this occasion.
What did they say?
Kilmarnock manager Alex Dyer: “We worked hard, especially second half. We showed them a little bit too much respect because they’re a good side and they’re top of the league.
“I don’t think we deserved to win but did enough to try and get a point. If we play like that, we’ll do alright.”
Rangers manager Steven Gerrard told Sky Sports: “I thought we were very efficient. Big three points, big win. We stood up, we competed. We defended extremely well. We could’ve scored more goals.
“To be fair to Kilmarnock, they always stay in the game. We had to stand up to that. We’re in a good place, we’re playing well. It’s too early to be getting ahead of ourselves. Really happy and proud of where we are right now but tonnes and tonnes of work to do.”
Kilmarnock visit St Johnstone on league business on Friday (19:45 GMT). The previous day, Rangers meet Benfica in Portugal (17:55), then play the last Premiership match before the international break next Sunday at home to Hamilton Academical (15:00).