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The away side had the better of the opening stages and should have taken the lead through Luca Toni, but the veteran stumbled as he took his shot and put it wide.
Toni had another chance after holding off the defenders in the area, but his strike on the turn went just wide of the post.
Verona eventually took the lead just after the half-hour mark through Bosko Jankovic, who saw his cross from the left evade everyone in the box and find the bottom corner.
Verona made it 2-0 on 43 minutes when Toni found Romulo with a cross to set him up for a simple finish.
Toni then made it three in injury time after he was found by Juan Iturbe, who left the Italian with a simple tap-in.
The away side continued to dominate in the second half and came close with an effort from distance from Iturbe, though his shot went just wide of the post.
Livorno pulled a goal back on 72 minutes through Paulinho, who struck into the bottom right corner from the edge of the 18-yard box.
Livorno continued to push for an equaliser, but struggled to create any clear-cut chances as they pegged Verona back.