The Reds are now tied for third place with Manchester United (a game in hand) and Aston Villa with 58 points each, eight more than sixth-placed Brighton. Crystal Palace are 13th with 43 points.
Liverpool had three shots on target and both were on target, unlike their rivals who had seven shots on goal but only scored once.
It was the 35th minute when Alexander Isak gave the home team the lead with his first goal of the year.
Just five minutes later, Liverpool's lead was doubled by Andrew Robertson, who is leaving Anfield at the end of the season.
Crystal Palace halved the deficit in the 71st minute through Daniel Munoz, while today's excellent Frederick Woodman was lying on the grass injured. The home team protested, but in vain.
The guests had several chances to reach a draw, and then Florijan Virc removed all doubts about the winner with a beautiful goal in the sixth minute of stoppage time.
Tottenham celebrated after six matches and four and a half months, as they were better than long-time write-off Wolverhampton Wanderers at Molineux (1:0)
The decisive goal was scored by reserve player Joao Palinha in the 82nd minute.
Despite the long-awaited triumph, Roberto de Zerbi's team remained first below the line, two points behind West Ham, who defeated Everton in London (2:1).
All three goals were seen in the second 45 minutes.
The Hammers took the lead with a goal from Tomaš Souček in the 52nd minute, but the Caramels equalized through Kiernan Desbury-Hall two minutes before the end of regulation.
It seemed that the teams would share the points, but reserve Callum Wilson made sure that didn't happen with a goal in the third minute of stoppage time.
Jerrod Bowen assisted on both goals.
