Both the Reds and Aston Villa will be fighting for points at Anfield this weekend.
The former defender has said that, though Brendan Rodgers' changing of formation was encouraging against Norwich City in the recent 1-1 draw, Liverpool have to make sure they earn the victory that would be expected of top four contenders.
"Failing to beat Norwich at Anfield means next week's home game against Aston Villa becomes very important," he said on BBC Sport.
"I hate to use the term must-win but it will be no good if the performance is better than the result again in that game, because Liverpool have to take three points.
"If they want to make the top four, Liverpool should absolutely be winning both games - if they don't beat Villa then the Merseyside derby at Goodison Park is next up and the pressure really will be on."
First off, Villa have what is sure to be a tempestuous Second City derby on Tuesday night in the League Cup, while the Reds have what on paper should be a much simpler home tie against Carlisle United.
Liverpool made a strong start to the season by recording two wins from their opening matches, followed by a draw away to Arsenal. Considering the Reds had kept clean sheets in their first three matches, there was every expectation that Brendan Rodgers had successfully addressed defensive issues from last season, however letting in three against both West Ham and then Manchester United suggested this was not the case, and Liverpool failed to win their third successive Premier League game when Norwich secured a 1-1 draw at Anfield on Sunday.
With the Merseyside club down in 13th position in the early stages of the league, and with captain Jordan Henderson ruled out for up to two months with a foot problem, it is no surprise that Lawrenson has identified the match with Tim Sherwood's side as a vital one for Liverpool.
Villa have had a number of narrow losses themselves this season, and will be determined not to slip into a relegation battle, having only narrowly escaped last time round. The Midlands club have won two out of the last three matches against the Reds, including a 1-0 win at Anfield last September, and Sherwood will be encouraging his players to believe that now is a good time to face their opponents, especially with doubts over the fitness of former star Christian Benteke.