Luca Antonelli's 10th-minute goal was the difference between the two sides at the Armando Picchi Stadium.
Livorno had several chances to equalise but were denied by a couple of fine saves from Genoa goalkeeper Mattia Perin.
The outcome is Genoa's first win on the road since November and has lifted the Grifoni to 10th in the standings.
Livorno have remained in the relegation zone and one point adrift of Chievo, who hold the last position of safety.
The visitors went into the game on the back of a 1-0 loss at Sampdoria, their first defeat at home since September.
Genoa showed great determination from the start and went ahead when Livorno goalkeeper Francesco Bardi failed to hold onto Alberto Gilardino's shot.
The ball fell to Antonelli and he struck it into the back of the net.
Livorno should have equalised in the 20th minute but Paulinho's goal-bound attempt was kept out by Perin.
Perin made another save four minutes later when he cleared Marco Benassi's 12-yard shot to make sure Genoa went into the break with a narrow advantage.
Livorno continued to press forward after half-time and Benassi and Djamel Mesbah saw their long-range efforts kept out by Perin.
In the 67th minute, the hosts wasted a good opportunity when Benassi fed Leandro Greco in the box and he struck the ball over the bar with only Perin to beat.Genoa defended deeply and held on for three points.