The England Under-21 striker scored his 10th goal of the season with a superb long-range strike which proved the only goal of the game and edged the Clarets closer to the top six and condemned Palace to their 14th away defeat of the season.
Eddie Howe`s men could have led inside the second minute when winger Ross Wallace collected possession 25 yards from goal and drove a shot just wide.
But moments later Burnley did strike the decisive blow when Rodriguez was afforded the time and space to strike a right-footed shot into the top left-hand corner from 25 yards.
Source: PA
Source: PA