Rodriguez met a pinpoint left-wing cross from Dwight McNeil with a perfectly angled glancing header to put the Clarets ahead in the 73rd minute.
Former Southampton striker Rodriguez had gone close to a first-half opener when he had a shot cleared off the line by Craig Dawson after Matej Vydra had struck the post.
Rodriguez had the ball in the net earlier, heading in an Ashley Westwood cross, but the effort was ruled out for offside.
The nearest Watford came was a second half effort from Troy Deeney which was cleared off the line by McNeil.
Watford are in 16th place, a point above the relegation zone.
Meanwhile, 11th placed Burnley move to 42 points.
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