On-loan Coventry midfielder Danny Swanson ended the champions' eight-match winning run in the league with a spectacular 54th-minute volley.
Ronny Deila's men remain six points clear at the top of the table, while Tommy Wright's St Johnstone climb back into the top six.
"It was a big setback. It is disappointing, but it's happened now and we have to learn from this and it's four days before we can make it good again," Deila told BBC Radio Scotland.
"Nothing is finished and we make it difficult for ourselves. This is not good enough.
"There were some tired players out there, but we should be better as a unit."
The success was just St Johnstone's second in their last eight league matches, and their first Scottish Premiership triumph at Celtic since August 2011.
"The goal was worthy of winning any game. It was a tremendous strike," Wright told BBC Radio Scotland.
"It shows a lot of character and belief in one another."
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