The Clarets named an unchanged eleven from the team that got thumped by Manchester City.
Watford made two changes to their 1-1 draw with Leicester City. Danny Welbeck and Tom Cleverley came in for Abdoulaye Doucoure and Roberto Pereyra.
Watford were the team with it all to play for but it was the Clarets who were the quickest out of the blocks.
Rodriguez forced Ben Foster into a smart early save, striking a dipping half-volley towards goal.
On his first start at Turf Moor, Josh Brownhill would have been eager to mark the match with a goal but his header from Dwight McNeil's pinpoint cross was blocked.
Burnley kept piling on the pressure and couldn't believe their lack of luck later on.
Foster was caught in no-mans land as Matej Vydra raced onto a raking ball over the top. Vydra's first-time chip bounced off the post but the chance remained alive.
The ball rebounded to Rodriguez who struck the ball past Foster but Craig Dawson made a brilliant goal-line clearance to preserve his team's clean sheet. Again, the ball landed to a Claret shirt but Brownhill badly screwed his attempt wide. Such misfortune was reminiscent of a bad FIFA game!
The visitors started to get a foothold in the match and Dawson went close with a header just before the interval. Still, the first-half was one that Burnley had the better of.
Nigel Pearson's men improved dramatically after the break. A tactical change created some fresh imeptous.
The Hornets had lots of territorial play but they found some openings from set-pieces.
Nick Pope spilled a corner from Will Hughes but reacted quickly to make a sprawling block as Craig Cathcart prodded an effort towards goal.
Troy Deeney is always a focal point for his team and the striker almost found the opener from Doucoure's corner but his header was blocked on the line by McNeil.
The ball rebounded nicely for Hughes but the midfielder smashed his effort over the bar from 12-yards.
Like Burnley, Watford were unable to show efficiency while they were in the ascendancy. And naturally, as the narrative often goes, they were punished.
McNeil found space on the left-hand side, whipping a delicious ball into the six-yard box. Rodriguez was the quickest to react, using the pace off the cross to guide the ball across Foster and into the far corner.
Burnley: Pope 7; Lowton 6, Tarkowski 8, Mee 7, Taylor 7; Brownhill 6, Cork 6, Westwood 7, McNeil 7; Rodriguez 8, Vydra 7.
Subs used: Pieters 6.
Watford: Foster 7; Femenia 6, Dawson 7, Kabasele 6, Masina 6; Capoue 7, Hughes 7; Sarr 7, Cleverley 5, Welbeck 5; Deeney 8.
Subs used: Cathcart 6, Doucoure 6, Chalobah n/a, Gray n/a.
Jay Rodriguez: A very energetic performance and he showed signs of a good understanding with Vydra. Unlucky not to score in the first period and his winning goal illustrated excellent opportunism.
Troy Deeney: A real nuisance. Holds the ball up really well but also has good awareness of others. Link-up play with Welbeck was promising but it just lacked a cutting edge.
Subs used: Cathcart 6, Doucoure 7, Chalobah n/a, Gray n/a.
Hope for the Hornets
Watford were poor to begin with but showed their metal after the break. They have a very athletic team and, while Deeney doesn't have the energy of his team-mates, his physical attributes are so difficult to handle. The skipper is also a clever footballer, linking up well with team-mates when the ball is on the floor, and is far from simply a big lump.
In comparison to others at the bottom, there is a bit more about this setup. Pearson has made them a more resilient unit and often looks to play percentage balls when attacking. The risk-free attacking style makes them less vulnerable to transitions and, with Deeney, they have a man who thrives on physical tussles.
The attitude from the Burnley players was tremendous. They were straight out of the blocks and could have been two goals ahead before the break.
Watford bounced back after half-time but the effort and endeavour remained constant. While their opponents could tweak their line-up tactically with substitutions, Burnley had to hope that their diligent approach would be enough. They needed a bit of luck and they got it.
Dyche replaced an injured Rodriguez with left-back Erik Pieters, a sign of just what Burnley are up against at the moment. All the more praise should go to the players for putting off-the-field politics to one side and knuckling down.
Watford will be on their own turf on Sunday, welcoming Southampton.