Barca suffered a shock 2-1 home defeat to Alaves last weekend, but despite a bright start from the hosts there was to be no repeat of that upset once Suarez teed up Messi to open the scoring after 15 minutes.
Suarez and Neymar tapped home to give Barca a healthy half-time lead before a Messi penalty and wonder strike from Rafinha made it 5-0.
Gabriel Appelt Pires's fine free-kick pulled a goal back for Leganes 10 minutes from time.
"We started very well. If we had started sluggishly, thinking we just needed to turn up to win, then we would have struggled a lot more," said Barca boss Luis Enrique.
Also Read
Messi, Suarez and Neymar started together for only the second time this season and, just as in a 7-0 thrashing of Celtic on Tuesday, showed no mercy as Leganes's high defensive line was torn apart in the first-half.
The hosts were given a warning when Diego Rico blocked Messi's goalbound effort from Barca's first counter-attack.
However, Leganes were punished moments later when Suarez showed great strength to burst down the right and his low cross was easily turned home by Messi.
The roles were reversed for Barca's second when Messi was gifted possession inside the Leganes half and waltzed past two defenders before teeing up Suarez for a tap-in.
Leganes did pose a threat going forward as Venezuelan winger Darwin Marchis caused Javier Mascherano, playing in an unfamiliar role on the right of Barca's defence, no end of problems.
Suarez bagged his second assist of the half just before the break when he was released clear on goal and selflessly squared for Neymar this time to roll into an empty net.
All of Barca's front three were involved for the fourth 10 minutes after the break as Neymar latched onto Suarez's ball over the top and was dragged down by Unai Bustinza.
Messi showed no signs of the penalty yips he suffered last season and at the Copa America in June as he fired into the top corner for his eighth goal in as many games this season.
And Leganes got the reward their display merited 10 minutes from time when Gabriel curled a sensational free-kick past the helpless ter Stegen.
Disclaimer: No Business Standard Journalist was involved in creation of this content