The Leith side have been humiliated by Malmo with a 7-0 Europa League defeat, lost 1-0 to Motherwell and fought back to claim a point with a 1-1 draw in their three Easter Road fixtures so far.
But they finally secured their first three points of what has been a torturous season to date with a 2-1 win in Ayrshire.
It came three days after chairman Rod Petrie gave his manager his vote of confidence through a statement issued on the club's website.
Fenlon, however, was adamant that only wins and not press releases will keep him in his job and stressed they must improve on their record in the capital.
"People keep asking me about the chairman's statement," said the Irishman. "I meet him every week. I don't need the club to tell me they have backed me since I've been here.
"All I need to do is to continue doing my job. It's a fantastic football club and I'm delighted for the supporters. It's not nice when your team is losing.
"Football is about results and unfortunately it's a horrible game when you are not winning matches. That's the way it is.
"We have won today and got ourselves off the mark. Now we have got to build on that. We have got to make sure we make Easter Road a difficult place to go to and try to win some matches there."
And the former St Johnstone midfielder secured the Hibees' long-awaited triumph when he outmuscled Darren Barr to get on the end of Lewis Stevenson's long ball before lobbing Craig Samson in the home goal.
Fenlon was delighted for the scorer, who has not escaped flak himself since moving south from Perth in the summer.
He said: "I was told this week that Liam can't play on the left. But he is a good player.
"I'm delighted for him. He has come here with a reputation for scoring goals but it hadn't started too well for him. Today will be a big confidence boost for him."
Killie boss Allan Johnston was disappointed with his side's lapse in concentration as they allowed another game to go by without securing victory.
Like Hibs, they went into the game without a win to their name following defeat to Aberdeen and draws with St Johnstone and St Mirren.
But Johnston is refusing to worry yet, saying: "We can't afford to give cheap goals away. We were in a great position again after Barry scored a wonderful strike but we give the lead away by losing a goal a minute and a half after half-time, which is just criminal.
"It's disappointing and we have got to improve in the final third.
"We're still trying to bring in one or two - a centre-half and a striker or wide player - and hopefully that will happen before the end of the transfer window."