The defender scored the only goal to put Atletico in command ahead of the second leg.
Atleti took the lead just four minutes before the break as Godin thumped a header home from Koke's superb right-wing cross. Neither side was able to find the net in a dismal second half, although Atleti will be happy to have avoided conceding an away goal.
However, it will face a stern task in the Basque Country in Wednesday's second leg, with Bilbao having won all but two of its games at the San Mames since its opening in September.
The 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 new year and saw chances pass him by yet again.
Both sides came close early with Costa failing to turn home Koke's cross from the left from 2 yards out and Ander Herrera blazing over for the visitors.
Koke was central to everything good about Atleti's play and provided a telling contribution, whipping a magnificent cross in that gave Godin the simple task of nodding past Bilbao goalkeeper Iago Herrerin.
Bilbao was neat in possession but toothless in the final third against the hosts' resilient defense. 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 halftime substitute Gabi failed to play him in on goal as Atletico launched a counterattack.
Though the 25-year-old was unable to break his six-game scoring drought, it ultimately did not cost Simeone's side, which is well-positioned to make the semifinals.