After going 68 league games unbeaten at Anfield, the defending champions were stunned 1-0 after Ashley Barnes' 83rd-minute penalty.
The Reds' run started with a 0-0 draw against Southampton and featured 55 wins and 13 draws, scoring 176 goals and conceded just 44 in the 68 games.
They also managed to keep 36 clean sheets, creating a fortress that helped propel them to a first league title since 1990 last season.
Liverpool are second on the all-time list for top-flight unbeaten runs at home, behind Chelsea's record of 86, which ended in October 2008.
But they have struggled for consistency this season and the loss to Burnley left them in fourth in the table, six points adrift of leaders Manchester United.
Liverpool are without a goal in their past four Premier League games, their longest such run since May 2000 (five games), taking 87 shots without netting, while it is the first time a Klopp side has gone four league games without scoring since his Mainz team did so in the Bundesliga, between November and December 2006.