Athletic had moved to within four points of fourth spot with a dramatic 3-2 win over Levante in mid-week but they barely created any clear chances in a highly physical encounter, only managing one shot on target.
Getafe's Spain striker Jaime Mata missed two clear chances to score either side of Rodriguez's strike but the one goal was enough for the home side to take all three points and their players embraced each other at the final whistle.
The victory took fourth-placed Getafe on to 50 points after 31 games although Sevilla climbed up to fifth in the standings on 49 points after beating Real Valladolid 2-0 away from home on Sunday with late strikes from Roque Mesa and Munir El Haddadi.
Valencia dropped to sixth place on 46 after their surprise 2-0 defeat at Rayo Vallecano on Saturday, with Alaves one point and one place below them after conceding a stoppage-time equaliser to draw 1-1 at home to Leganes on Sunday.
Athletic are eighth on 43.
(Reporting by Richard Martin; Editing by Toby Davis)
