Three goals in the final 15 minutes of the first half from Bosko Jakovic, Romulo and Luca Toni put the visitors firmly in control, 3-0 up at half-time, but Livorno made a fight of it as Paulo Betanin and Leandro Greco scored two in the space of two minutes with less than 20 minutes to go.
The win keeps Verona firmly in the hunt for European football, but does nothing to help Livorno's bid to get out of trouble at the wrong end of the Serie A standings.
Verona were well worth their half-time lead after an opening 45 minutes that was one-way traffic in the direction of Livorno's goal.
Marco Marquinho and Juan Iturbe both had early sighters before Francesco Bardi made a good save to deny Toni in the 19th minute.
The goalkeeper was busy again moments later, turning over a cross-cum-shot from Michelangelo Albertazzi, and he then did well to turn a thunderbolt from Marco Donadel over after Toni had shot narrowly wide.
An increasingly inevitable goal arrived after 33 minutes when Jankovic's attempted cross found its way into the net even without Toni making contact as he lunged at the ball.
Bankovic and Romulo both went close again before the Brazilian added the second in the 44th minute, turning in Toni's cross from close range.
There was time for a third before half-time as Iturbe set up Toni to score into an open goal in stoppage time.
Livorno were much improved after the break with Ramos Emmerson and Innocent Emeghara having early efforts on goal.
At the other end, Romulo curled a shot just wide and Bardi tipped over Toni's shot but the momentum was now with Livorno.
Rafael De Andrade made a great save to keep out Betarin's strong header but the goal was coming and arrived in the 72nd minute.
Betarin beat Donadel to break into the area and then slotted a shot just inside the left-hand post.
And seconds later they had a second, with Betarin turning provider to set up Geco for a close-range finish.
However, they could not find an equaliser, with a long-range effort from Emmerson which drifted over the best they could come up with.