Struggling Bilbao, who have held Real Madrid and Barcelona to draws this season but have not won since the opening day, took the lead in the 36th minute when Inaki Williams nudged the ball across the line after team mate Mikel San Jose had rattled the underside of the crossbar.
Williams missed a glaring chance to double Athletic's lead early in the second half and was left ruing it when Thomas Partey sent a long range pile-driver into the net to level in the 61st minute.
Williams quickly replied again to restore Athletic's lead, producing an ice-cool finish after racing on to a through ball by close friend Iker Muniain but Atletico midfielder Rodri levelled in the 80th minute by scoring his first goal for the club with a glancing header from a corner.
Godin, who remained on the pitch only because Atletico had made the maximum three substitutes, hobbled into the box to contest a late free kick and sent the home fans wild by heading in the winning goal, initially ruled out for offside then awarded after consulting the video assistant referee (VAR).
The win follows Atletico's impressive 2-0 victory over Borussia Dortmund in the Champions League and takes Diego Simeone's side second in the La Liga standings, one point behind Barcelona who host Real Betis on Sunday.
"We could still feel the effects of our efforts from the other day (against Dortmund); they pressed us well but we battled until the end, we put up a fight and we got our reward," said Godin, who scored the goal which clinched the La Liga title for Atletico in 2014.
"I hurt myself in Athletic's second goal, I couldn't keep running. I was going to go off but there were no changes left so I stayed on. Simeone told me to stay up front as a striker."
Athletic's double goalscorer Williams added: "We feel terrible, we had the win in our hands but we go home feeling empty."
Valencia built on Wednesday's vital Champions League win over Young Boys by grabbing only their second victory in La Liga this season, beating Getafe 1-0 with a penalty from captain Dani Parejo.
(Reporting by Richard Martin; editing by Clare Fallon)
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