Liverpool finally lost after 10 straight games without defeat
AFC Bournemouth scored the first goal when Evanilson netted after 26 minutes, with an assist by Alex Scott, making it 1-0.
Dominik Szoboszlai equalised for Liverpool with an assist by Mohamed Salah, in the 80th minute.
The result means the fifth game in a row without a win for Liverpool.
It leaves AFC Bournemouth in 13th place and that Liverpool are in fourth place.
It had been a long time since Liverpool finished a Premier League game on the losing side. But Saturday's away match to AFC Bournemouth saw the end of their unbeaten run of 10 games as the match finished 3-2.
In the 33rd minute Álex Jiménez increased the lead for AFC Bournemouth with an assist by James Hill.
Liverpool's Virgil van Dijk made it 2-1 in the 45th minute as he found the net, with an assist by Dominik Szoboszlai.
The deciding goal came in extra time, when Amine Adli scored for AFC Bournemouth.
AFC Bournemouth play Wolverhampton Wanderers on 31 January, Aston Villa on 7 February and Everton (away) on 10 February. Liverpool's next opponents are Newcastle United on 31 January, Manchester City on 8 February and Sunderland (away) on 11 February.