Klose's goals came in the space of seven minutes as he first profited on a Antonio Candreva assist before the Livorno defence was compliant in allowing him to effectively kill off the game inside half and hour.
The veteran Germany striker, who had just two league goals coming into the game, could have had a hat-trick too after he had hit a post shortly before opening the scoring.
Lazio had not tasted league success since they beat Cagliari 2-0 seven games ago - the last time Klose had scored - although Livorno have now taken just a single point in five games to drop them into the relegation places.
Signs that Klose would rediscover his goalscoring touch against the strugglers were evident early on.
The 35-year-old fizzed a shot over in the seventh minute and moments later headed inches past the left-hand post after running on to a Senad Lulic cross.
Klose then deflected Candreva's long-range effort flush against the post. It was not surprising then when he did open the scoring on 19 minutes.
Candreva was the architect as his delivery set up Klose to fire home from close range.
Seven minutes later it was 2-0 and Klose had doubled his season's tally.
Livorno's defence had to take some of the blame after a poor clearance left Klose with a task of scoring from a similar distance to his first.
He did so and the game was almost up for a Livorno side that has failed to score in four of their past five games.
Sergio Betanin Paulinho at least showed signs of life before the half-time whistle, shooting over before then directing a header over from Andrea Coda's cross.
After the break Lazio's lead was not unduly threatened with Klose's best chance of completing his hat-trick coming when he headed over a Stefan Radu cross on 49 minutes.
Federico Ceccherino and substitute Innocent Emeghara were both off target when Livorno did muster up a forward thrust, while Candreva drew a late save from Francesco Bardi that would have made it 3-0.