The Yellow Submarine finished fourth in La Liga last season but have stumbled out of the blocks this year, winning just one of their opening six league games.
Having overseen the departure of star playmaker Santi Cazorla to big-spending Malaga in the summer, Roig is prepared to give Garrido time to get things right.
He told the AS newspaper: "We have the best coach for this team.
"Garrido is a great coach and a club man. Last season was spectacular and this year will not be any less so. I have full confidence in its possibilities.
"Things are not as bad as some are suggesting. We are a healthy club who does not owe anything to anybody.
"We have had a bad start to the season but things have been exaggerated."
Aware that his club cannot compete financially with the likes of Real Madrid and Barcelona, or even Atletico Madrid or Malaga, Roig accepts they have to take a different path.
"Two seasons ago I said we must adjust the squad for what was to come," he added. "We have reduced the budget and sold key players.
"We knew the team could suffer but we know this is the road we have to go down.
"In two or three seasons the team will be healthy and competitive. It will take time to make this team one of the best again."