Jurgen Klopp won his first match in the Premier League since taking the helm at Anfield, with Philippe Coutinho firing twice and Christian Benteke grabbing the third against the current champions.
The victory leaves Liverpool just eight points off leaders Manchester City and four from a Champions League qualification spot.
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And Lawrenson was impressed by the Reds' performance at Stamford Bridge and feels the win will give their rivals plenty to think about.
"There was a real statement made by Liverpool at Stamford Bridge," the Match of the Day pundit wrote in his column for the Liverpool Echo.
"And it was up there on the scoreboard for all to see, and for all watching at home to take in.
"Coming off the back of Klopp's first win as manager against Bournemouth last Wednesday, it's clear the feel-good factor has returned to Anfield.
"After 40 minutes, though, you were fearing the worst.
"We had plenty of possession, but we were playing in front of Chelsea without really threatening in the final third.
"But it is amazing how a moment of genuine quality can change a game. And what a goal from Philippe Coutinho to provide it.
"The pressing game worked really well for us at Stamford Bridge.
"Of course, the system needs a bit of refining. We've already seen at home that when opposing teams sit back and defend, the pressing game isn't always that, well, pressing."
Klopp's philosophy is slowly starting to fizzle through at Anfield.
A real sense of belief has been reinstated since the German took charge and each individual member of the Reds squad looks fully committed to their new manager.
Under Brendan Rodgers, heads dropped when Liverpool went a goal down and it was rare they fought their way back against weaker opposition, nevermind a team like Chelsea.
However, the Merseyside outfit dominated following Ramires' opener for the Blues after four minutes and a piece of individual class from Coutinho levelled things up.
Since Klopp took charge of Liverpool, he's struggled to have strikers fit and relied on Divock Origi to lead the front line in his first four matches.
The 20-year-old, however, was left at home at the weekend and Roberto Firmino started as the Reds' number nine in west London.
The Brazilian struggled for the most part before Benteke came off the bench with 36 minutes to go when the scores were level.
The Belgium international made the difference, first assisting Coutinho's second goal and then wrapping up the three points with his fourth goal for Liverpool since his summer move from Aston Villa.
Klopp will be hopeful he can have Benteke fully fit for the remainder of the season and Lawrenson believes it is also key.
"Roberto Firmino started up front and while he was often involved, he didn't really get in where he could hurt Chelsea," the 58-year-old added.
"Everything changed, though, the minute Christian Benteke emerged from the bench in the second half.
"He instantly brought presence to the attack. We knew that if we put the ball up there, it was going to stick.
"For the rest of the team, they know there will be an end product to all their build-up work and that makes such a difference, especially to midfield players."