Innocent Emeghara scored twice for the visitors, once from the penalty spot, with Leandro Greco and Paulinho also scoring.
The impressive Simone Zaza responded for Sassuolo but could not stop them from slipping to the foot of the embryonic table.
There was tentative start and it was not until the ninth minute that Greco tried his luck from outside the box but Antonio Rosati made a smart save.
Emeghara then made surging run down the right and found himself in space but his shot from 25 yards went well over the crossbar.
The home side had their first chance after 21 minutes but Zaza's speculative effort went wide of Francesco Bardi's left post.
Bardi was made to do his first meaningful piece of work just after the half hour as Simone Missiroli supplied a great cross to Karim Laribi who got a good head to it but the keeper was on hand to make the save.
Sassuolo had worked their way back into the game but they were stunned a minute before the break as Greco hit a beautiful curling free-kick which gave Rosati no chance.
Eusebio Di Francesco's men were clearly keen to get back in it straight after the break and Jasmin Kurtic's shot from 30 yards was well gathered by Bardi.
Zaza was then just wide with a close-range effort before Sassuolo got the goal their positive play merited.
Zaza again at the centre of things as Laribi provided the cross and the striker headed home in the 67th minute.
But just two minutes later Livorno regained their lead as Greco found Emeghara who burst forward and provided a cool low finish.
It took just six minutes for it become 3-1 as Paulinho beat Reto Ziegler and gave Rosati no chance from 12 yards.
The scoreline looked harsh on the newly promoted side and it was about to get worse as Rosati fouled Andrea Luci in the box and Emeghara scored the resulting spot kick.