The defender scored the only goal of the game to put his side in the driving seat in the Copa del Rey quarter-final
Atleti shaded an even first half at the Vicente Calderon and took the lead just four minutes before the break as Godin thumped a header home from Koke's superb right-wing cross.
Neither side were able to find the net in a dismal second half, although Atleti will be happy to have avoided conceding an away goal.
However, they will face a stern task in the Basque Country in next Wednesday's second leg, with Bilbao having won all but two of their games at the new San Mames since its opening in September.
Victory was just Atletico's second in five games and coach Diego Simeone will undoubtedly be concerned about Diego Costa's ongoing woes in front of goal.
The forward has not found the net since the turn of the year and saw chances pass him by yet again.
Both sides went close early on with Costa failing to turn home Koke's cross from the left from two yards out and Ander Herrera blazing over for the visitors.
Koke was central to everything good about Atleti's play and he provided a telling contribution, whipping a magnificent cross in that gave Godin the simple task of nodding past Bilbao goalkeeper Iago Herrerin.
Bilbao were neat in possession, but toothless in the final third against the hosts' resilient defence. Oscar de Marcos looked to test Thibaut Courtois from 25 yards on the hour, but his driven effort bobbled wide.
Costa was visibly frustrated as half-time substitute Gabi failed to play him in on goal as Atletico launched a counter-attack.
Though the 25-year-old was unable to break his six-game scoring drought, it ultimately did not cost Simeone's side, who are well-placed to make the semi-finals.