Referee Bobby Madley pointed to the spot in the final minute after Dale Stephens was adjudged to have fouled Jese at the back post from a Xherdan Shaqiri cross.
However, Mathew Ryan kept out Charlie Adam's spot-kick, before Lewis Dunk prevented Adam getting the rebound. From the resulting corner, Ryan kept out Adam's low cross before Anthony Knockaert headed a Mame Diouf effort off the line.
On the penalty incident, Stephens said: "I think it was soft. It was an unbelievable save and a great follow up by Dunky as well. I’d got my arm out but my eyes were on the ball. I’d got a little hand just to stop him running around. It happens – it happens every game now.
"If they’d scored the penalty I would be very disappointed now. We probably rode our luck a bit towards the end but I think we probably should have been out of sight first half and in the second we created some good opportunities.
"I think it’s probably a good point overall considering the result last week."