There was no repeat of the drama from when the two sides met in March, and shared eight goals, but there were still enough chances for Dortmund to win the game, fuelling Klopp's malaise.
"We had the shining golden opportunities," he said. "We could even have had more than just three big chances.
"And (Stuttgart goalkeeper) Sven Ulreich was sensational.
"We just have to live with it that a draw in a home game feels like a defeat. We would have deserved to win it."
The draw sees Dortmund slip to 11 points behind league leaders Bayern Munich while it extends Stuttgart's unbeaten run to five games.
"I think the result is fair," said Stuttgart coach Bruno Labbadia. "We fought for this point.
"At times, we had the luck you need to survive in Dortmund."