Mohamed Salah sent Liverpool hearts racing when he opened the scoring after just three minutes with a touch of pure class.
The Egypt international jumped on a loose ball and hit a difficult shot into the top right corner.
It was an end-to-end game with both sides missing key opportunities to distance themselves from the other.
It looked like Steve Bruce's side had nicked a point on 93 minutes when Wilson bundled the ball into the net but replays showed it had bounced off his arm.
That wasn't to be the last action of the game, though, as Liverpool failed to clear a long ball and Willock shot it past Alisson to rescue the point.
The result could be crucial in Liverpool's fight to secure Champions League football. They are in sixth place, a point behind Chelsea and West Ham United who are in fourth and fifth respectively and play each other later on Saturday.
For Newcastle, the draw puts them nine points clear of the relegation zone and in 15th position.
Klopp said after the game that Liverpool didn't deserve a Champions League spot as they did not fight hard enough on the day.
"In a specific way, we don't fight enough," he said.
"I didn't see us that we deserved today to play Champions League next year. But we will see what we can do. In a few days we will play Manchester United which will be a tough one as well. But we learn... or we don't play Champions League. That's it."