Sigurdsson proved Swansea's matchwinner as he pounced on a poor defensive clearance from Ragnar Klavan, the Iceland international securing Paul Clement's maiden win and Liverpool's first league loss at home since January 2016.
Adam Lallana and Tom Carroll had the best chances of a lacklustre opening half, but that all changed after the restart as the game burst into life.
Llorente struck his first goals away from the Liberty Stadium to put Swansea 2-0 ahead before Roberto Firmino lifted some of the gloom with his seventh in the league this season.
And the Brazilian dragged his side level with 21 minutes to play with a fine finish as Swansea's defensive frailties reared their head once again.
A dramatic second half had one final twist, though, as Sigurdsson ensured a crucial three points that lifts Swansea out of the bottom three, while leaving Liverpool seven points adrift of Chelsea having played a game more.
Given their imperious home record it was no surprise to see Liverpool dominate possession in the early stages and Emre Can should have got them off the mark from Jordan Henderson's ninth-minute cross.
Liverpool continued to press and Lallana saw an overhead kick clear the crossbar from Firmino's cross, the Brazilian having superbly spun away from Alfie Mawson.
Despite the hosts' control of proceedings it was Swansea who came closest to breaking the deadlock as Carroll’s deflected cross thudded against Simon Mignolet’s right-hand post.
It was a rare threat on the Liverpool goal, though, and the hosts resumed control leading up to the break but a resolute Swansea were able to get crucial blocks in to frustrate them and the Anfield crowd.
And the visitors built on their foundation three minutes into the second half as Llorente poked home from six yards after Federico Fernandez's header had been controlled by Wayne Routledge.
The Spaniard then doubled his tally moments later with a towering heading from Carroll's sensational left-wing cross.
Anfield was not silent for long, though, as Firmino halved the deficit to end a crazy seven-minute spell with a bullet header of his own after Martin Olsson had misjudged James Milner's cross.
It came as no surprise that Liverpool duly equalised after 69 minutes as Firmino got in on the act again, this time the forward chesting a Georginio Wijnaldum cross before lashing past Lukasz Fabianski as Olsson and Jack Cork watched on.
However, the game's next key moment was something of a surprise as Sigurdsson ghosted in at the back post to slide home Swansea's third.
Desperate for another equaliser Liverpool piled the pressure on in the closing minutes, with Fabianski saving from Divock Origi and Dejan Lovren before a late melee saw Lallana's looping effort hit the crossbar as Swansea clung on to their first Premier League win at Anfield.