The home side were staring at just their second defeat in all competitions this season before the returning Diogo Jota rescued a point at Anfield.
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 early but the complexion of the contest changed dramatically soon afterwards when Liverpool defender Andy Robertson was sent off.
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.
"I don't think I could have asked for more from my players, especially after going down to 10 men," Slot told Sky Sports.
"I couldn't have asked for more in terms of performance and results.... At the start of the game, you always want to start intense.
"After 30 seconds, one of ours was on the floor and needed treatment for a while so we couldn't get the intensity more."
Slot has enjoyed a sensational start to his managerial reign at Anfield, with his side topping the Premier League and the Champions League.
But in their previous top-flight outing they conceded a late equaliser to draw 3-3 at Newcastle and they have now dropped more points.
Enzo Maresca's in-form Chelsea will hope to take full advantage when they host Brentford, though Liverpool will have a game in hand at the end of the weekend.
Red card
Former Manchester United midfielder Pereira opened the scoring at the Kop end in the 11th minute, 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, with Gakpo producing a diving header for 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 in the 76th minute.
But after Harvey Elliott and Darwin Nunez had both sent efforts wide, Jota, who had only been on the pitch for a few minutes, proved his long injury lay-off had dulled none of his predatory instincts by beating Bernd Leno from the middle of the penalty area.
Alisson's save from Adama Traore protected the point and with Everton doing their neighbours a favour by drawing at Arsenal, Liverpool moved to 36 points -- stretching their lead at the top over Chelsea to five points, at least for 24 hours.
Arsenal are a point further back.
