But the dropped points -- for the second game in a row -- mean Chelsea can close the gap at the top of the table to just two points if they beat Brentford at home on Sunday.
Andreas Pereira put the visitors ahead in the 11th minute and six minutes later Liverpool were reduced to 10 men when defender Andy Robertson was sent off.
But Arne Slot's men came out strongly after half-time and were level in the 47th minute when Cody Gakpo headed home Mohamed Salah's inswinging cross.
Rodrigo Muniz appeared to have won the game for Fulham in the 76th minute but the returning Jota, playing his first match since October, levelled 10 minutes later.
Former Manchester United midfielder Pereira scored the first goal of the game at the Kop end, meeting Antonee Robinson's cross on the stretch to beat goalkeeper Alisson Becker.
Robertson had been culpable for the opener and his attempt to recover from a poor touch succeeded only in bringing down Harry Wilson, which referee Tony Harrington deemed to be a denial of a goalscoring opportunity.
Liverpool failed to get back on level terms in the first half despite their bold approach but two minutes into the second period they were level when Salah cut in from the right and curled a cross to the far post where Gakpo scored his seventh goal in 10 matches.
Salah drove wide in a four-on-two attack and that miss looked pivotal when substitute Muniz deflected in Robinson's cross 14 minutes from time.
But after Harvey Elliott and Darwin Nunez had both sent efforts wide, Jota proved his long injury lay-off had dulled none of his predatory instincts by sending Bernd Leno the wrong way from the middle of the penalty area.
Becker's save from Adama Traore protected the point and with Everton doing their neighbours a favour by drawing at Arsenal, Liverpool stretched their lead at the top over second-placed Chelsea to five points.
But Enzo Maresca's in-form team can cut that lead to two points if they beat Brentford at Stamford Bridge.
© 2024 AFP