Maxi grabbed his first goal 31 seconds into the Premier League game before the Argentine struck again in the seventh minute. Dirk Kuyt then made it 3-0 when he scored for the fifth successive match.
Shell-shocked Fulham pulled one back in the 57th minute through Moussa Dembele but Maxi, who also recorded a treble against Birmingham City on April 23, slammed in a fantastic long-range shot 13 minutes later and Luis Suarez added a fifth.
Steve Sidwell scored late for Fulham but it did not change anything for Liverpool (58 points) who climbed above Tottenham Hotspur into the Europa League spot in fifth position before Spurs (56) travel to fourth-placed Manchester City (62) on Tuesday.
"It was just one of those days for us," Liverpool defender Jamie Carragher told Sky Sports. "We've been playing well of late, confidence is good, we are looking forward to games."
While Suarez's nippy runs and deft touches were at the heart of many Liverpool surges, Fulham's Australian keeper Mark Schwarzer had much to answer for as the visitors went three up in 16 minutes.
Schwarzer kicked Suarez's deflected cross into the path of Maxi for the opener and did not cover his near post well enough for the second but his worst blunder was for the third goal when he let Kuyt's soft effort slip past him.
He could not do anything about Maxi's third, a stunning strike from 25 metres, while at the other end Sidwell's consolation goal into the top corner was of a similar class.
Liverpool's win came the day after arch rivals Manchester United closed within one point of winning the league to stand on the brink of breaking the record they share with the Anfield club of 18 English top-flight titles.
Since Kenny Dalglish took over the reins at his old club in January, Liverpool have turned their season around and have in the past month recorded two 3-0 wins and a 5-0 victory.
"We have done fantastically well since January, that's great credit to the players and... the backroom staff," said Dalglish. "We'll just keep going and see what happens."
A miserable first half of the season had put them close to the relegation zone and qualification for the Europa League would salvage something from the campaign and increase the already loud calls for Dalglish to be offered a permanent contract.
The Scot was tight-lipped over whether talks about his future were ongoing, saying: "When we've got something to tell you, we'll tell you".
While a Champions League place is still a mathematical possibility, Liverpool would need Spurs to win in Manchester and for the results in the final two matches of the season to go their way.
