Godin pulled a muscle in the 64th minute as Inaki Williams gave Bilbao a 2-1 lead at the Wanda Metropolitano, but Atletico's captain stayed on as they had made all three of their substitutes.
Rodrigo headed them level with 10 minutes left, before Godin's close-range finish, initially ruled out for offside, was allowed to stand by referee Jose Sanchez, after consultation with his video assistant referee.
It proved a spectacular finale to a pulsating encounter, in which Bilbao led twice after Williams' opener was cancelled out by a long-range strike from Thomas Partey.
"During the Williams goal I felt my muscle go," Godin said afterwards. "Simeone told me to stay on up front, in case a corner or a ball came into the area. I tried to help and it worked out well."
Simeone added: "It is one of the most exciting victories in the history of this stadium."
Victory means Atletico move to within a point of Barcelona, who play at home to Real Betis on Sunday. Bilbao remain 17th, without a win since the opening day of the season.
That run has, however, included draws against both Barca and Real Madrid, and for much of this contest they were the better side too.
But after beating Borussia Dortmund in the Champions League on Tuesday, Atletico sustained momentum with a comeback made all the more impressive by their lengthy list of absentees.
Godin and Diego Costa both returned but 19-year-old Francisco Montero had to play alongside Godin in central defence, with Lucas Hernandez, Jose Gimenez and Stefan Savic all out injured, along with Thomas Lemar and Koke in midfield.
"We knew it would be a difficult game, with many players missing, and after a big week for us," Godin said. "But we will be energised a lot by winning like this. It lifts the fans, and us players too."
Atletico started brightly enough but Bilbao never looked uncomfortable and gradually grew into the game.
Markel Susaeta fired wide, after Rodrigo carelessly conceded possession, and then instigated the goal, darting free of the sleeping Filipe Luis.
Williams troubles Atletico
Susaeta squared for Mikel San Jose, whose first-time shot was brilliantly clawed onto the post by Jan Oblak. But the ball rolled agonisingly across the line, where Williams was perfectly placed.
It was the first league goal Atletico had conceded in a first half this season and there were whistles as the players departed at the interval.
Thomas lifted the mood with a booming effort from range that flew past Iago Herrerin and into the corner. But, level in the 61st minute, Atletico were behind again in the 64th as one pass sent the speedy Williams clear.
Godin clambered at the striker's heels but could not recover. Williams slid in his second.
Atletico pushed again and Bilbao never looked safe sitting back. It was a simple corner this time, delivered by Thomas and glanced inside the far post by the head of Rodrigo.
There was still time for a winner and it was Atleti that snatched it. Saul Niguez kept the ball alive at the back post, Griezmann flicked towards goal and Godin poked it over the line.
The celebrations were halted by the linesman's flag. But only temporarily.
Earlier on Saturday, Valencia had also benefitted from VAR as they followed up their Champions League victory over Young Boys by winning 1-0 away at Getafe.
Daniel Parejo converted an 81st-minute penalty after Bruno Gonzalez's foul on Kevin Gameiro was spotted on review.
Bruno was sent off for a second yellow card as Valencia held on for only their second La Liga win of the season.