The victory, with Fabregas netting a hat-trick and Alexis scoring twice, restored Barca's 13-point advantage over Real Madrid at the top after their arch rivals had thumped Levante 5-1 in the earlier kick-off.
It was a timely confidence boost for Barca ahead of Wednesday's Champions League quarter-final, second leg against Paris Saint-Germain and included the emotional return of defender Eric Abidal a year after he underwent a liver transplant.
Coach Tito Vilanova was also back on the Nou Camp bench for the first time since he returned from a programme of cancer treatment in New York.
It is still not clear if Messi, who damaged a hamstring in last week's 2-2 first-leg draw in Paris, will be available to face PSG.
On Saturday's evidence Barca would manage without him as Fabregas and Alexis combined superbly to see off Mallorca, who will end the weekend at the bottom of the table.
Fabregas played a one-two with Alexis and slotted past goalkeeper Dudu Aouate to open the scoring in the 20th minute.
Alexis followed up to net the rebound from a Fabregas shot two minutes later and another one-two between the pair led to a second for Fabregas in the 37th minute.
Alexis completed his double two minutes after that when Fabregas clipped a pass over the Mallorca defence and the Chilean took one touch before poking the ball into the corner.
Fabregas scored his third in the first minute of the second half when he ran on to a superb Andres Iniesta pass and a festive night for the fans was completed when Abidal returned to a rousing ovation as a 70th-minute substitute for Gerard Pique.
ACROBATIC VOLLEY
Gonzalo Higuain, Kaka and substitutes Cristiano Ronaldo and Mesut Ozil were all on the scoresheet for Real in their win at home to Levante.
Coach Jose Mourinho left a number of key players out of his starting line-up as Real prepare to defend a 3-0 lead at Galatasaray in Tuesday's Champions League quarter-final, second leg.
Mid-table Levante took a surprise lead in the 31st minute when Michel finished off a swift attack before Higuain levelled with an acrobatic volley five minutes later.
Kaka put Real ahead from the penalty spot in the 38th minute after Christian Lell's handball and Ronaldo volleyed into the net from close range and set up Ozil for Real's fourth after coming on in place of Jose Callejon at half-time.
Ozil stroked in Real's fifth in added time as the champions moved on to 65 points from 30 matches.
"The players who do not get many chances have these matches to prove they are capable of being in the side," assistant coach Aitor Karanka told a news conference.
"We are also motivated by finishing second, because Atletico are close behind and we have to keep going so they can't go ahead of us," he added.
EUROPEAN HANGOVER
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