Southampton started brightly as Adam Lallana linked well with Rodriguez early on, but the forward failed to hit the target with his effort from the edge of the box.

The visitors continued to look the most threatening side at Selhurst Park, but they were forced to wait until the 37th minute before they could break the deadlock.

Jay Rodriguez of Southampton celebrates as he scores their first goal during the Barclays Premier League match between Crystal Palace and Southampton at Selhurst Park on March 8, 2014 © Getty Images

Rodriguez took full advantage of a mistake from Jason Puncheon and beat Palace goalkeeper Julain Speroni in a challenge before converting into an empty net from his position on the floor.

The Saints were inches away from being further ahead when Lallana headed the ball to Rickie Lambert, but the striker could only hit the post from close range.

Yannick Bolasie of Crystal Palace takes on Jack Cork of Southampton during the Barclays Premier League match between Crystal Palace and Southampton at Selhurst Park on March 8, 2014 © Getty Images

Palace attempted to increase the intensity of their play in the second half, but they struggled to compete with Southampton as the visitors continued to justify their advantage.

Lambert came close to doubling his side's advantage with a curling free kick that almost troubled Speroni, who was later forced to keep out a low strike from Luke Shaw.

Palace piled on the pressure late on as they went in search of an equaliser, but Southampton held firm to secure all three points from their away trip.