The Toffees entered the match having not beaten their rivals in the Premier League since October 2010, while they were looking for their first top-flight victory over the Reds at Anfield since September 1999.
Richarlison made the breakthrough early on before Sigurdsson scored from the penalty spot in the 83rd minute as Liverpool lost four home league matches in succession for just the second time in their history.
Everton are now level on points with their sixth-placed opponents in the Premier League table and sit just three points off fourth-placed Chelsea with games in hand over both teams.
The Toffees did take the lead in the third minute, though, when James Rodriguez played a clever pass to Richarlison, and the Brazilian took one touch before firing the ball into the far corner of the net.
The game's next attempt on target also came from Everton in the 17th minute, but Rodriguez's long-range free kick never threatened to trouble Alisson Becker between the home side's sticks.
Liverpool's first real chance arrived shortly after as Roberto Firmino found himself free inside the Everton box, only for his effort to take a deflection off Michael Keane and drop behind for a corner; Jordan Pickford then had to scramble across his goal to keep out a brilliant Jordan Henderson volley in the next attack.
Pickford made another smart save in the 21st minute to keep out a long-range Trent Alexander-Arnold effort as the home side started to turn the screw after a frustrating opening to the contest.
Liverpool had to make a change in the 30th minute, though, as captain Henderson limped off the field to be replaced by Nat Phillips as the team's injury problems at the back worsened.
Everton had a brilliant chance to double their lead in the 33rd minute when Seamus Coleman met a cross from Digne, but Alisson was on hand to make an excellent save to keep out the header.
Curtis Jones then sent an effort wide of the post down the other end, but Liverpool could not find a leveller before the interval as the half-time whistle blew with Richarlison's early strike separating the two sides.
Klopp resisted the temptation to make any changes at the interval, while Everton were also unchanged at the start of the second period, but fit-again Dominic Calvert-Lewin went through a warm-up at the break.
Mane had a half-chance in the early stages of the second half when he met a cross from Andrew Robertson, but Pickford was on hand to keep out the attacker's header, before the Senegal international headed another cross from the same side over the crossbar minutes later.
Everton were struggling to get out as the pressure from the hosts continued, and Pickford had to keep out a low effort from Thiago Alcantara in the 56th minute to keep his side ahead.
Sigurdsson tested Alisson with a strike in the 64th minute as Everton started to offer more of a threat in the final third, with Calvert-Lewin following the Iceland international off the bench for the visitors.
Liverpool had a huge chance to level the scores 20 minutes from time when Mohamed Salah broke into the box before attempting to poke the ball past Pickford, but the England goalkeeper made another smart save.
Firmino then struck one over the crossbar in the 73rd minute as Everton continued to hold strong, with the Toffees looking to overcome the Reds at Anfield in the Premier League for the first time since 1999.
Everton were then awarded a penalty in the 81st minute when Calvert-Lewin was brought to the ground by Alexander-Arnold, who tripped up the striker when he looked favourite to slot past Alisson; the referee was sent over to the monitor for a second look, but it did not take long to award the spot kick.
Sigurdsson stepped forward for the Toffees, and the attacker picked out the bottom corner to double the away side's advantage in the 83rd minute of the contest.
Firmino had a low strike deflected wide of the post before Pickford kept out an effort from Georginio Wijnaldum as the home side looked to answer back, but it was not to be for Klopp's side, who have now suffered seven league defeats this season.
Liverpool's title challenge has evaporated in recent weeks, and the Reds are now in a serious battle for a top-four position, with Chelsea and West Ham United both above them, while Everton are level on points with a game in hand.
The reigning champions are next in action away to basement side Sheffield United on February 28, while Everton will be looking to build on the result when they host Southampton on March 1.
LIVERPOOL: Alisson; Alexander-Arnold, Kabak, Henderson (Phillips 30'), Robertson; Thiago (Origi 87'), Wijnaldum, Jones (Shaqiri 63'); Mane, Firmino, Salah
EVERTON: Pickford; Coleman, Holgate, Keane, Godfrey, Digne; Doucoure, Davies, Gomes (Sigurdsson 59'); Rodriguez (Calvert-Lewin 62'); Richarlison (Iwobi 86')