Former Gers striker Kris Boyd scored his 20th goal of the season in the 54th minute to take the Ladbrokes Premiership points back to Ayrshire.
It was the seventh home league defeat of the season for the Light Blues - the most since 1914/15 - and the home side heard boos of frustration at the final whistle.
Second-placed Rangers are just two points ahead of Aberdeen who have a game in hand with leaders Celtic nine ahead.
Asked what his message to the fans was, Murty said: “Still believe. Keep believing in us. I have got a changing room in there 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.
“We know that and there is lots to play for this season. We need to get them energised and believing to the end of the season.
“You go in there and 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.”
Informed of the unwanted home record, Murty said: “We know we haven’t been good enough at home this season.
“That was encapsulated today where a team was very organised and made it difficult for us to play and capitalised on us making a mistake.
“That seems to be a theme of this season and that is one that we are very keen to break.”
Rejuvenated Kilmarnock shrugged off the midweek William Hill Scottish Cup quarter-final defeat after extra time and penalties to Aberdeen to go six points behind fourth-placed Hibernian.
Boss Steve Clarke said: “That was the seventh game in 21 days which was a big ask for the players.
“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.
“I don’t take any credit, I give credit to people I work with.
“My staff are excellent and my players have been magnificent and the crowd were great as well.
“There maybe wasn’t so many here today but you could hear them shouting and screaming in the background and getting a little bit excited - and trust me when your two brothers are Kilmarnock fans you know how excited they are getting at the moment.”