First-half goals from MSN -- the first time the trio have scored in the opening 45 minutes away from home -- set the tempo, before the result was sealed after the break thanks to Messi's second and a Rafinha bullet.
Leganes did get a consolation, as Gabriel bent a superb free-kick around the wall and out of the reach of Marc-Andre ter Stegen.
It means Barcelona has now scored 12 goals in five days, after its 7-0 demolition of Celtic in the Champions League.
"We made use of the space and we were switched on from the start," manager Luis Enrique said after the match.
"The tactical set up did have its risks but we faced up to it well and I am proud."
Barcelona was looking to avoid a second consecutive defeat to a promoted club, after a shock 2-1 defeat against Alaves at the Nou Camp last weekend and Messi settled any nerves inside 15 minutes.
Suarez clumsily bundled his way down the right, before passing the ball square to Messi for the Argentine to turn in at the near post.
It was the first La Liga goal ever scored at Leganes' Municipal de Butarque stadium and Messi's 34th at a different ground, surpassing Real Madrid legend Raul's previous tally of 33.
Up until the opening goal, Leganes had more than held its own against the reigning La Liga champion, but the Barcelona front three find scoring easy enough without chances being presented to them on a plate.
A loose pass in midfield found its way to Messi, who ghosted past two defenders and unselfishly passed the ball to Suarez to tap into an empty net.
With their tails up, Messi, Neymar and Suarez began to purr and it wasn't long before the lead was three.
Messi played a through ball to into the path of Suarez, who raced clear on goal and squared to Neymar to delicately cushion the ball over the line -- MSN scoring in the perfect order.
With the result secure, it took only 10 minutes of the second half for Barcelona to extend its lead.
Neymar sped in behind full-back Bustinza Martínez, who couldn't live with the Brazilian's pace and dragged his opponent to floor to concede a blatant penalty.
Messi stepped up and expertly hammered the spot kick into the top corner.
Another loose ball in defense gifted Barcelona its fifth, this time allowing Rafinha to cut inside onto his left and curl an outstanding shot into the top corner.
The loudest cheer of the afternoon, however, was saved for 10 minutes before the end.
Arda Turan hauled Gabriel down on the edge of the box, before the Brazilian picked himself up, dusted himself off and curled his free-kick around the wall and into the bottom corner.
A deafening roar went up from the 10,958 capacity crowd, seemingly forgetting -- or just not caring -- about the five-goal deficit, as those present witnessed the first La Liga home goal in the club's history.
Despite a flurry of late chances, neither side was able to add to the score sheet and victory ensures Enrique's men move level on points with Real Madrid.
However, Zinedine Zidane's side has a game in hand and travels to Espanyol Sunday, although it will be without the services of its two key men, Cristiano Ronaldo and Gareth Bale.
The Welshman picked up an injury in Tuesday's Champions League win over Sporting and Ronaldo has the flu.