Lennon led St Mirren to Scottish League Cup glory last season but they have won just three Scottish Premiership matches in 2013 and produced a poor display at Easter Road.
But Lennon said: "If I did not believe that I was the man for this job, I wouldn't be sitting here.
"If that decision is taken away from me and I'm no longer required at this football club, then my question would be: What is it that they want the new man to come in and achieve?
"Do they want them to come in and make sure the team survives? I've already done that every season I've been here.
"Do they want successful cup runs? I've been in four quarter-finals and won a national cup.
"Or is it to simply finish higher up the league. I've already given them their highest SPL finish ever.
"There is no guarantee a new manager would even match that, never mind improve on it. I certainly believe that, given time and my own ability, I will improve on that.
"I believe in the character of my group. Danny Lennon has hand-picked every player in that dressing room.
"It was myself who handed out the contracts which allow them to go and pay their bills. so there is no doubt in my mind that the players are still with me."
Fenlon happy with Hibs
James Collins and Paul Heffernan were on target for the hosts, who have now gone five games without defeat after making a slow start to the season.
Manager Pat Fenlon said: "I'm delighted, I thought it was a real team performance. We were very solid and created some good chances with some good football.
"All footballers are the same. If you win games, you gain some confidence and momentum and it just breeds from there. You can see it in training every day.
"All we have done is transferred that from the training pitch to here.
"You can't buy confidence. It's about working through the hard days, which we certainly had at the start of the season. It's about believing what you are doing is right and being bold and brave enough to say we will get it right."