Chances were few and far between for much of the opening period at Selhurst Park on Saturday, but it was a mistake by a former Southampton player that saw the deadlock broken.
Jason Puncheon applied a weak header to a long ball out of the visitors’ defence, with Rodriguez pouncing to get the better of Julian Speroni and poke the ball into an empty net.
The win takes ninth-placed Southampton to 42 points, one better off than at the end of last season, while Palace are now just two points clear of the relegation zone.
Tony Pulis rewarded Glenn Murray for his game-changing cameo at Swansea City last weekend by handing the striker just his second Premier League start, but it was frustrating opening 10 minutes for the hosts as they fell foul of the offside flag on three occasions.
Southampton took control of the game by having the lion’s share of possession, but neither side were giving goalkeepers Speroni or Artur Boruc particular cause for concern.
Rodriguez got a sight of goal shortly before the half-hour mark, but Damien Delaney slid in well to ensure the Argentinian remained untested, before Southampton were forced into an early change when Jack Cork went down with an apparent ankle injury - Victor Wanyama his replacement.
The game finally sparked into life eight minutes before the break when Puncheon conceded possession with a weak defensive header, forcing Speroni to rush out of his area to challenge Rodriguez.
The England international came out on top and subsequently rolled the ball into an empty net with the game’s first shot on target, taking him level with Luis Suarez, Daniel Sturridge and Wayne Rooney on eight away goals for the season.
Another of the visitors’ World Cup hopefuls Rickie Lambert very nearly doubled their lead on the stroke of half-time, but prodded his effort against the right-hand post from just inside the 18-yard box, before Yannick Bolasie drilled a powerful strike straight at Boruc at the other end.
Palace came from a goal down at the break to earn a point at Swansea last Sunday, and Bolasie tried to force the issue early in the second period, but was brought down by Dejan Lovren 20 yards out as he surged goalwards, Mile Jedinak blazing the resulting free kick well over.
The second 45 minutes followed much the same pattern as the first – Southampton enjoying the majority of possession, but failing to look especially dangerous.
Luke Shaw, who picked up his first England cap against Denmark on Wednesday, tried his luck from 20 yards out in the 73rd minute, but it was an easy save for Speroni.
Having started the half brightly, Palace never really looked like finding an equaliser, and Southampton were comfortable as they held on to the three points.
