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Lallana struck in the 95th minute to seal a 5-4 win moments after Sebastien Bassong had equalised for the Canaries.
Klopp was swamped by his players - and his trademark specs slipped off.
"I have a second pair of glasses, but I can't find them," said Klopp. "It's hard to find glasses without glasses."
Klopp has broken his glasses before during a match, with that pair residing in his former club Borussia Dortmund's museum.
"We won for the first time against Bayern Munich and Nuri Sahin broke my glasses. Today it was Adam Lallana," he added.
Read about how the dramatic game unfolded
Klopp may have been smiling at half-time but just moments earlier he was fuming at his players for allowing Bassong to make it 4-4.
The Reds' defending had been particularly poor throughout, with Norwich having scored from every shot they had on target to lead 3-1.
"There was about 70 minutes that were really good," said the 48-year-old.
"Conceding four goals is in general not good. Conceding from set plays is the biggest rubbish ever because it is too easy for the other team.
"Usually we concede from a corner but now it was from the second ball - we were like rabbits in the headlights, but we won the game in the end."
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Klopp was not the only person who felt his side should have defended better in the game.
"I am not happy obviously," said Norwich boss Alex Neil.
"The game was in our hands at 3-1 but after that we managed to turn a really good performance into a bit of a shambles."
Liverpool manager Jurgen Klopp and his players celebrated Adam Lallana's stoppage-time winner at Norwich so wildly that the German ended up breaking his glasses.