The England Under-21 forward scored his 10th goal of the season with a superb long-range strike to edge the Clarets closer to the top six and condemn 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 left-foot shot just wide.
Moments later Burnley struck the decisive blow when Rodriguez was afforded the time and space to strike a right-foot shot into the top left-hand corner from 25 yards.
Palace sought to find an immediate riposte but David Wright fired narrowly over the crossbar from distance and midfielder Darren Ambrose then saw a shot blocked by Clarets defender Michael Duff.
Wallace then fired over from 20 yards as both sides looked threatening when on the attack.
Rodriguez should have doubled his tally in the 14th minute when he headed wide and and three minutes later Wallace wasted a fine opening when he blazed over the crossbar from just six yards.
Steffen Iversen then tried his luck from 12 yards in the 20th minute but Burnley defender Tyrone Mears cleared the danger.
Burnley defender Daniel Fox had a shot blocked by Patrick McCarthy and Rodriguez shot over the crossbar as the hosts looked to make their dominance pay.
They created another fine opening in the 29th minute when an inswinging corner from Wallace was met by Rodriguez but his shot from 12 yards flew over the crossbar.
In the 38th minute Jack Cork - who scored Burnley's winner in last weekend's 2-1 victory at Preston - headed goalwards but Palace goalkeeper Lewis Price made a fine save to avert the danger.
Chris Iwelumo then headed at Price from 12 yards and Chris Eagles tried to give Burnley a killer second goal in the 51st minute when he drilled in a 25-yard strike which Anthony Gardner did well to block.
In the 59th minute Burnley boss Howe replaced Iwelumo with Wade Elliott due to injury.
Moments later Wallace should have scored but sliced a shot wide from 12 yards out as Palace continued to lead a charmed life.
Fox tried his luck from 30 yards but Wright made a fine block to avert the danger.
Wallace then shot wide before Ambrose had a 25-yard free-kick tipped over by Lee Grant.
McCarthy headed over the bar and then Elliott headed wide as Burnley held on for a potentially priceless three points.