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The Reds' home undefeated record in the Premier League had extended back to April 2017. But they lost on Thursday, thanks to Ashley Barnes' late penalty, awarded after the player himself had been brought down by Alisson.
Liverpool are now winless in their last five league games (three draws, two defeats) scoring just one goal. The champions had chances in the first half - with Divock Origi hitting the woodwork - but Burnley were dogged throughout.
In the second half, Nick Pope pulled off some fine saves. Then after 83 minutes, Barnes escaped Fabinho's clutches and ran through on goal, only to be brought down by Liverpool's keeper. Barnes nervelessly slotted home the pen.
Liverpool had 26 chances by full-time but failed to take any of them. And Klopp lost his rag in the same game his team lost their long unbeaten record.
Klopp has cut a frustrated figure of late and that boiled over when the game was still 0-0, as he had a confrontation with Dyche inside the tunnel while the two managers went off for half-time. The LFC boss could be seen yelling and gesticulating wildly - while Dyche stood his ground and shouted back.
The pair almost looked set for a physical coming together, until Dyche - eyeballing Klopp - walked back to the away dressing room.
What Dyche might've said to spark that reaction is open to interpretation. But the incident on the pitch that caused Klopp to temporarily lose his temper was clear.
Fabinho and eventual goalscorer Barnes had a slight coming together on the pitch, with the Liverpool man shown a yellow card. It didn't look spectacular - but Andy Robertson was clearly angry with Barnes for going down and referee Mike Dean paused while VAR checked the incident.
No further action was taken but it was tense on the pitch during the check - and more than tense in the tunnel as both Klopp and Dyche got into it on their way back for their half-time team talk.
However it was Dyche who had the (gravelly) last laugh on Thursday as his Clarets side pulled off a big upset. The result leaves Liverpool six points off leaders Manchester United halfway through the season, while Burnley moved seven points clear of the relegation places.