The League Cup winners played very well towards the end of the first half and took the lead through Esmael Goncalves but they were less impressive after the break, and were hanging on for a point somewhat after Kris Boyd had equalised.
Lennon said: "Kilmarnock came out for the second half and squeezed the game a lot higher, and stopped us moving the ball as well as we had been.
"We have got to handle that side of things better. We made poor decisions in terms of the possession we had.
"But it was still a well-earned point with some beautiful football in the first half and plenty of character, commitment and sweat to see ourselves through.
"It's great, because the boys are showing a positive mentality right now.
"Before the cup final we went up to Dundee, went 1-0 up but missed chance after chance, and Dundee came back out and turned it round.
"But tonight we showed that other side to dig in and hold on to a valuable point."