Barcelona, who left Lionel Messi on the bench, are already three points off the pace, after losing two of their first five matches.
Suarez said the defeat is a concern, adding that winning such games is what guarantees the league title.
“It’s a worrying defeat, a defeat that hurts. And a defeat that we have to take a real good look at, because there are lots of things that we have to improve,” he told AS.
“There’s a long way to go yet, but these are the types of games that win you the title. We have to be tougher if we’re going to win the league. We have a tough year ahead of us.”
Victory against the Granada side they have beaten 11 times in their last 12 league meetings would have taken the Catalans top of the table on goal difference.
But despite the introduction of Lionel Messi at half-time, it was the home team who took their opportunity to leapfrog Sevilla and Atletico Madrid to the summit.
The hosts went ahead in the second minute through Azeez. Roberto Soldado dispossessed Junior Firpo of the ball, fed Antonio Puertas and his cross was headed in by the striker from close range, leaving Barca trailing for the fourth time in five league matches.
Granada doubled their advantage in the 66th minute. Vadillo converted from the penalty spot, following a handball from Arturo Vidal, who had just come on as a substitute.
“You can’t go behind so early,” Suarez further noted.
“That made everything that bit tougher for us. We had possession of the ball, but we couldn’t make it count. We have to analyse what happened.”
Barcelona have lost on the road at Athletic Bilbao and Granada already this season, as well as drawing at Osasuna and Borussia Dortmund, and have scored only two goals.
“What we have to look at is the way we’re playing as the away side, because the lack of cutting edge and chances is worrying,” Suarez also said.
“We’re Barcelona, and we have to go out there to win the game – we can’t start the way we did.”