The hosts looked to be heading for all three points through a Kenny McLean goal but in a remarkable late turnaround, conceded twice in the last 10 minutes to fall to a fourth defeat in five games.
The result will intensify the pressure on Lennon's position, but he said afterwards: "We know things need to improve. But there has been many a good manager that has been through the roller-coaster ride I'm currently on and come through at the other end.
"But we need a bit of patience. Football is an emotional game but there have been clubs that have shown patience in similar situations. Look at St Johnstone last season. They were in the same exact position we are in now - one point from four games.
"But they showed patience and Stevie Lomas took them to their highest ever SPL finish and qualifying for Europe.
"Am I worried? I'm concerned. I'm concerned (given) the standards we have set at this club that we go and produce what we did today. We were a shadow of our former selves.
"I spoke to the chairman there and the board have been absolutely fantastic with me. Top drawer. If my journey at this club was to end tomorrow, I would have no regrets. It has been a pleasure to come here and work.
"But it's something I don't want to give up easy. I love a challenge. The part of the journey I'm going through is not nice but it is one I will relish and I will put it in my tool bag and carry it along with me on my hopefully long managerial career."
Partick boss Alan Archibald was understandably delighted, adding: "I'm a wee bit surprised that we have found our feet so quickly but I'm just pleased we have started so well. It gives us confidence.
"Our fans turned out in their numbers again today, so you can see we bring a lot to the Scottish Premiership.But one thing we want to do is to get away from the cuddly-toy image that we are tagged with."