Morata scored twice but chipped a glorious chance straight into the hands of Wayne Hennessey when through on goal in the final minute.
The Spaniard had opened the scoring with a clever turn and shot after 32 minutes, before Andros Townsend equalised with a low right-footed drive in the 53rd minute.
Eden Hazard then made an immediate impact on his return to the side after coming off the bench.
The Belgian’s free-kick was fired home by Morata to make it 2-1 just after the hour mark and Pedro converted Marcos Alonso’s cross with 20 minutes left on the clock.
But Morata admitted the miss was still playing on his mind despite the positive result.
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“I can score a hat-trick so it was not the perfect day but we will take the three points,” he said.
“That is the most important thing for us. We want to fight until the end for the Premier League.
“We played good today. I think a lot about the hat-trick but I am improving and I want to keep helping my team to win.
“Palace score to make it 1-1 so we knew we needed to go and win it and stay at the top with the other teams. We needed the three points.
“It is very hard because the Premier League is the most competitive league in the world. It will be very difficult.
“We want to take all the points. It is the only way to try to fight for the league.
“When you have a bad period, it takes long to recover. Last year was difficult for me for many things.
“You don’t expect an injury but now I am good, it is nice to be scoring.”
Maurizio Sarri’s unbeaten side move above Liverpool on goal difference and into second place.
They are just two points behind leaders Manchester City, who thumped struggling Southampton at home earlier on Sunday.