Griezmann lashed home a brave Diego Costa knockdown in the 34th minute but Portu snatched a shock equaliser for the away side 17 minutes from time, meaning that Atletico stay second and move up to 43 points, five ahead of Valencia, who can draw level with Atletico if they win at Las Palmas later.
Villarreal moved above Real Madrid and knocked Zinedine Zidane's side out of the top four by beating Levante.
The draw at the Wanda Metropolitano means that Barca can move 11 points clear with a win at Real Betis on Sunday and almost end discussion over the La Liga title.
Costa made his mark again on Saturday, with his second assist since returning from Chelsea adding to his three goals in five games in all competitions.
He lay down in a heap after being clattered by Girona keeper Bono as he headed on for Griezmann, but it was the stopper who came off worse and had to be substituted at half-time.
Fifth-placed Sevilla drew level on 32 points with Real Madrid after crushing Espanyol -- who beat Barcelona in the Copa del Rey midweek -- 3-0 in a dominant display in Catalonia that gave Vincenzo Montella his first league win since arriving in Spain in late December.