Defender Van Dijk has an ankle injury, while midfielder Ward-Prowse and forward Rodriguez picked up knocks in Wednesday's EFL Cup win at Liverpool.
Midfielder Granit Xhaka is suspended for Arsenal following his red card against Burnley last Sunday.
Forward Theo Walcott is back in the squad following a calf injury.
Southampton will be without Virgil van Dijk, James Ward-Prowse and Jay Rodriguez for Saturday's FA Cup fourth round tie against Arsenal.