First-half goals from Charlie Adam, Kris Boyd and Nacho Novo put Rangers well in control and the victory was a tactical triumph for coach Paul Le Guen.
In torrential rain, Rangers made an uncertain start with goalkeeper Lionel Letizi looking particularly uncomfortable.
But the Glasgow side recovered and took the game to Livorno. Boyd headed wide after good work from Thomas Buffel before Adam saw a fiercely-struck shot blocked.
Rangers took a deserved lead after 27 minutes when Boyd headed a Steven Smith cross into the path of Adam and he blasted a left-foot volley past Marco Amelia from 20 yards.
It was 2-0 three minutes later when Nacho Novo was pulled down in the penalty area by Samuel Kuffour and Boyd sent Amelia the wrong way with the spot-kick.
Livorno grabbed a lifeline three minutes later when they were awarded the softest of penalties after Julien Rodriguez was adjudged to have fouled Cristiano Lucarelli.
Lucarelli took the penalty himself and struck his shot just wide of Letizi's reach.
But Rangers were not disheartened and scored again moments later. Boyd was again the provider with an excellent header and Novo did well to keep his right-foot shot down from the edge of the area.
Rangers continued their positive play in the second half and Adam swiped a left-foot shot just a couple of yards too high.
But Letizi had to be sharp to get his hand to a near-post header from Tomas Danilevicius to preserve Rangers' two-goal advantage.
Buffel was unfortunate not to score a fourth for Rangers after 64 minutes with a ten-yard volley which Amelia did well to block.
Livorno substitute Stefano Morrone then tested Letizi again with a well-struck shot which the goalkeeper pushed away for a corner.
The Italian side threw players forward in the closing stages and Lucarelli pulled a goal back in the final minute with a close-in shot.
Rangers substitute Filip Sebo hit the crossbar in injury-time as the Scots almost put the icing on the cake.