Juan Rodriguez broke the deadlock with an 11th-minute header and Vazquez put the visitors firmly en route to three points with a close-range finish just before the hour mark.
Mikel San Jose halved the deficit in added-on time but his strike came too late for Bilbao to push for an equaliser.
Marcelo Bielsa's men have now won just two of their opening nine La Liga fixtures and are hovering just above the drop zone following their third defeat in a week.
Bilbao had shown plenty of endeavour in last week's La Liga defeat at Valencia and their Europa League loss at French outfit Lyon.
And they started well tonight [Sunday] as Gomez Ibai gave Getafe keeper Miguel Moya an early sighter from distance.
But the hosts fell behind when Pedro Leon's ball into the area was brilliantly headed into the bottom corner by Juan Rodriguez.
Xavi Torres missed the target from close range as Getafe continued to press Bilbao while at the other end Aritz Aduriz headed over from Ibai's corner.
Abdelaziz Barrada then served a notice of Getafe's intent as his 25-yard rocket cannoned off the bar shortly before the interval.
Bilbao could not benefit from the let-off and they fell two behind in the 58th minute when Vazquez showed tremendous innovation with a volley past Gorka Iraizoz.
Bielsa knew he had to shake things up and brought on Fernando Llorente immediately after.
The change was nearly rewarded but the 27-year-old Spain international saw a header come back off the bar.
The hosts continued to press forward and Ibai's close-range effort brought a smart save out of Moya and although San Jose pulled one back after converting from Ismael Lopez's corner, it proved too little, too late.