Two Liam Craig goals turned it around at Rugby Park after Barry Nicholson had scored for the home side, leaving Fenlon to celebrate his first win since May in the week he was publicly backed by chairman Rod Petrie.
"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' 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."
Kilmarnock are yet to win, and their manager Allan Johnston said: "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."