The Brazilian broke the deadlock after just three minutes into the game, latching onto James Rodriguez's amazing pass and beating Alisson Becker to find the bottom corner.
Liverpool crawled their way back into the match by dominating possession and pushing Everton on the back foot.
However, clear-cut chances were hard to come by, and the visitors sealed the game through a contentious penalty from Sigurdsson in the 83rd minute after Trent Alexander-Arnold was penalised for a foul.
For only the second time in their history, and the first time since 2002, Liverpool have lost four home games in a row. They now trail runaway league leaders Manchester City by 16 points, who also have a game in hand.
On that note, here are the Liverpool player ratings:
Alisson Becker - 6/10
He made several important saves but could do nothing about either of Everton's goals. Alisson Becker is now without a league clean sheet in seven games.
Trent Alexander-Arnold - 6/10
It was another poor game from the Premier League's reigning young player of the year. Trent Alexander-Arnold also conceded a penalty in the second half.
Ozan Kabak - 6/10
His wait for a league victory continues, but Ozan Kabak put on an improved show, completing 93% of his passes, winning four aerial duels and making a game-high five interceptions.
Jordan Henderson - 7/10
The Liverpool captain was solid in defence once again but limped off with an injury just 30 minutes into the game. The reigning champions' defensive woes seem to have no end in sight.
Andrew Robertson - 5/10
Pretty much like his full-back partner Alexander-Arnold, there has been a significant drop in performance level from Andrew Robertson this season. The Scotsman struggled to link up with his attackers, and his crosses lacked venom.
Curtis Jones - 6/10
There were no derby heroics for Curtis Jones, who couldn't revive a hapless Liverpool midfield. He did make one good effort which curled just wide of the target, though.
Georginio Wijnaldum - 6/10
One of the few consistent starters for Liverpool, Georginio Wijnaldum, showed an excellent work-rate. He worked hard on and off the ball but lacked the end product.
Thiago Alcantara - 5/10
Thiago Alcantara has got a lot of stick lately. He can expect some more after his Everton outing, as the Spaniard was outclassed in midfield.
Mohamed Salah - 6/10
Mohamed Salah's three-game scoring run came to an end after a disappointing outing, as the Egyptian was firmly kept under wraps by the visitors. Salah's theatrics, however, will be remembered long after this match:
Roberto Firmino - 6/10
Roberto Firmino was closely marked by Everton and struggled to fully test the visitors. In fact, only one of his four attempts was on target.
Sadio Mane - 5/10
Sadio Mane showed plenty of promise in Liverpool's midweek win at Leipzig but came a cropper against Everton.
Ratings of Liverpool Substitutes
Nathaniel Philipps - 5/10
The centre-back hit a shot into the side-netting shortly after coming on. But Nathaniel Phillips couldn't much to stop Sigurdsson in the lead-up to the penalty.
Xherdan Shaqiri - 5/10
Xherdan Shaqiri was a spectator, to say the least.
Divock Origi - N/A
Divock Origi arrived too late to influence proceedings.