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.
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, while 11th placed Burnley move to 42 points.
"We were fortunate to be level at half-time, we didn't impose ourselves on the game. Our second half performance was much better," said Watford manager Nigel Pearson.
"It is a missed opportunity but no point in dwelling on these situations. They are good at what they do and saw the game out."
Burnley manager Sean Dyche said his team had shown strong spirit to bounce back from Monday's 5-0 hammering from Manchester City.
"Over seven and a half years we have built a culture and environment that behind it has a strong mentality. We bounced back after the other night," he said.
Dyche has been critical of the Burnley board for failing to resolve several player contract issues but shrugged off a media report suggesting he may leave the club.
"I am still here. I have had this before, still working with the players. It has been an emotional week. Someone wrote the first time in seven years I had questioned the board, I think that's pretty good," he said.