The Catalans started the match with the same line-up that dispatched Sevilla before the worldwide break with Suarez and Lionel Messi continuing to lead the line.
Two goals from Luis Suarez and Paulinho's late strike helped the Catalans extend their lead at the top of La Liga to seven points, although Valencia can recoup the difference with a victory against Espanyol on Sunday.
Alexander Szymanowski went close to equalising on 35 minutes with a right-foot effort from the left side of the area but Marc-Andre ter Stegen denied him with a fine save as he pushed the ball wide of the far corner at full stretch. The important thing was to win.
Suarez fired home the opener after 28 minutes before netting his second on the hour mark.
Garitano said on his club's website: "This season I hope to find a better version than the Leganes of last season and compete in these kinds of games, which would be a big step for us".
However, they still boast the third meanest backline in the division - only Barca (four goals) and Atletico Madrid (six) have conceded less - and Valverde admits Saturday's clash will be the start of a testing run of fixtures for his league leaders.
Atletico, Real light up Madrid — Laliga
And the Wanda Metropolitano side strengthened their hand when Griezmann agreed a new deal, doubling his release clause from £85million to £170m.
"Perhaps 0-3 is a bit excessive".
However, having thrown away the title last season with defeats at Malaga, Deportivo la Coruna, Celta Vigo and at home to Alaves, Valverde wants Barca's minds concentrated on Leganes.
"I wasn't anxious about scoring, I was just trying to feel good", he said after the game.
First-choice central defenders Gerard Pique and Samuel Umtiti are both just one yellow card away from being suspended for the trip to Valencia next weekend. I recognise the effort Leganes has made today despite the defeat.
The German goalkeeper was called into action again early in the second period when Leganes substitute Claudio Beauvue was too quick for the Barca defence only to fire straight at Ter Stegen.
However, Barca piled more pressure on the chasing Atletico and Real ahead of the Madrid derby and rubbed salt into Leganes' wounds when Paulinho rolled into an empty net when a loose ball broke off Messi in stoppage time.