Rangers manager Graeme Murty has urged the club's fans to "keep believing in us" after their seventh home loss of the season.
Kilmarnock inflicted a 1-0 defeat at Ibrox, following Celtic's 3-2 win at the home of their Old Firm rivals last weekend.
Rangers' record of losing seven of 16 home games is their worst since the 1914-15 season.
"We know we haven't been good enough at home this season," Murty said.
"I have a changing room that needs to keep believing. That appeared to be lacking on the pitch at times. But we need as a team to give the fans something to believe in."
Former Rangers striker Kris Boyd delivered the decisive blow for Kilmarnock, scoring his 20th goal of the season in the 54th minute.
The full-time whistle prompted booing from the home fans, with Rangers having won only seven of their home games this season, and drawing two.
The defeat leaves second-placed Rangers two points ahead of Aberdeen, who have one game in hand, with leaders Celtic nine ahead.
"We know there is lots to play for this season," Murty said. "We need to get [the players] energised and believing to the end of the season.
"If anything, they are more disappointed than me. They know that this place needs to be turned into a fortress. We haven't managed to do it.
"That seems to be a theme of this season and that is one that we are very keen to break."
Kilmarnock recovered from the disappointment of losing on penalties to Aberdeen in the Scottish Cup quarter-final replay at Rugby Park in midweek to beat Rangers for a second time this season.
The Ayrshire side also drew at Ibrox in the sides' other league meeting this term, and are now unbeaten in seven league games.
"That was the seventh game in 21 days, which was a big ask for the players," said Kilmarnock manager Steve Clarke.
"There was a bit of fatigue in the first 20 minutes, took us time to grow into the game and the last 15 minutes we were digging in.
"I haven't got a word which could praise this group of players enough. They find a way to get results.
"The run of wins we are on, the way we compete with the top teams, it has to be great for the Kilmarnock fans to watch us because as a coach I am really proud of the players and they should be really proud of their efforts as well.
"Trust me, when your two brothers are Kilmarnock fans you know how excited they are getting at the moment."