Suárez headed in a pass from Lionel Messi in the 53rd minute to cancel out Youssef En-Nesyri's opener. Substitute Vidal put Barcelona in front after a defensive error with 11 minutes left.
Leganés was close to taking a point against the defending champions until midfielder Rubén Pérez knocked a corner toward his own net, allowing an unmarked Vidal to tap in the winner.
"We pulled out a win at a very tough pitch against a team that is fighting for its life," Vidal said.
"We have to improve and keep teams from scoring against us early. But the important thing is that we took the three points and keep the lead."
The hard-fought victory kept Barcelona ahead of second-placed Real Madrid on goal difference.
Barcelona hosts Borussia Dortmund in the Champions League on Wednesday.
Barcelona coach Ernesto Valverde aligned his top four attacking players in a starting XI for the first time this season, with Ousmane Dembele and Antoine Griezmann flanking Messi and Suárez.
But Leganés in its second game under former Mexico manager Javier Aguirre stifled Barcelona in the first half with its two tight defensive lines packed in front of its box.
The hosts took a 12th-minute lead after Pérez passed to Roque Mesa, who went on to play the ball wide to En-Nesyri. The Morocco striker finished the move by curling home a left-footed strike that gave Marc-Andre ter Stegen no chance.
En-Nesyri has scored in three consecutive rounds, including a late equalizer to earn a 1-1 draw at Real Sociedad in Aguirre's debut.
The closest Barcelona came to scoring before halftime was a header from Suárez that goalkeeper Iván Cuéllar stretched to push wide.
The visitors improved after the break with Messi playing a key role.
Gerard Pique headed a corner kick from Messi off the post. Minutes later Messi earned a free kick from a foul outside the area. The Argentine found Suárez with a lob that Suárez headed under Cuéllar to start the comeback.
Valverde reinforced his midfield with Ivan Rakitic and Vidal. They replaced an ineffective Griezmann and Sergio Busquets, who had picked up a yellow card that will mean he is suspended for next round's trip to Atlético Madrid.
Vidal's goal was his fourth of the campaign.
Right back Moussa Wague made his first appearance of the season for Barcelona, which was without the suspended Sergi Roberto and injured defenders Jordi Alba and Nelson Semedo.