The Galicians will feel hard done by at just taking a point from a game in which they dominated possession and squandered a number of chances, and hit the woodwork twice.
Rafinha, the brother of Bayern Munich midfielder Thiago and who joined Luis Enrique's side on loan from Barcelona in the summer, broke the deadlock with just under half an hour gone.
Piti, who joined Granada on a free transfer from Rayo Vallecano, where he was the top scorer last season, levelled on 60 minutes.
Celta went into the game on the back of a morale-boosting victory at Real Betis and looked intent on maintaining their positive start to the season.
They had Lucas Alcaraz's side on the back foot for almost the entire first half, dominating possession and edging closer towards the goal.
Their pressure paid off when their high pressing forced Granada to surrender the ball in their own area, resulting in Rafinha shooting into an empty net.
Alex Lopez was inches away from doubling the lead moments later, striking the near post from the right hand side after a pass from Charles.
The hosts began the second half with even more ambition than they had shown in the opening 45 minutes and could have extended their lead on three occasions just after the break, with Lopez blazing over and Charles Dias de Oliveira failing to hit the target twice following crosses from Augusto Fernandez.
The Brazilian striker, who had scored in Celta's two previous games, had an even better chance just before the hour mark, but shot just wide after a through ball from Hugo Mallo.
He was made to regret that particular chance just two minutes later, when Piti ran into the box from the right hand side and shot past Celta goalkeeper Yoel Rodriguez.
Alex Lopez came close to restoring Celta's lead not long after when he hit the post for a second time, while Granada goalkeeper Roberto Fernandez thwarted Michael Krohn-Dehli in injury time.
Celta have now surrendered leads to draw in both of their home games this season but remain unbeaten, while Granada have notched up four points.