Premier League: Cardiff 0 Southampton 3
The Englishman took his tally for the season to nine in all competitions with a quick-fire first-half double at the Cardiff City Stadium as the visitors heaped misery on Malky Mackay.
The home fans protested against owner Vincent Tan before, during and after the match as uncertainty continues to surround Mackay's future, with things taking a turn for the worse on the pitch as Rickie Lambert completed an easy triumph for Southampton before half-time.
There was no way back for Cardiff in the second half and Mackay's side have now won just one of their last eight top-flight matches.
Cardiff were without Gary Medel through illness, with Fraizer Campbell deputising, while Southampton boss Mauricio Pochettino introduced Luke Shaw, Maya Yoshida and Jose Fonte into his back four after a less than convincing defensive display in the 3-2 loss to Tottenham.
Rodriguez tried to set the early tempo for the visitors with a dangerous run forward but was stopped in his tracks by Kevin Theophile-Catherine.
The hosts spurned a huge opportunity to take the lead in the 11th minute when Peter Whittingham failed to convert Craig Noone's cross with the goal gaping.
And they were made to pay for that miss when Rodriguez struck twice in six minutes to put Southampton in the driving seat.
The first arrived after 14 minutes. Adam Lallana sprung the offside trap before teeing up Rodriguez for a simple finish.
Lambert laid on the second with a pinpoint cross that England international Rodriguez turned home from inside the box without breaking stride.
And Lambert got in on the act himself before the half-hour mark with a close-range finish, capitalising on Cardiff's poor defending of a short corner.
Paulo Gazzaniga was called into action at the other end when Whittingham's shot deflected off Fonte, forcing the Italian into an excellent reflex save.
He almost undid that good work with a poor clearance a minute later but Campbell could not make the most of his chance.
Mackay introduced Andreas Cornelius in place of Peter Odemwingie at the break and the hosts came out fighting.
Campbell headed narrowly over from a corner and Cornelius' effort forced Gazzaniga into an unconvincing, scrambled save.
Rodriguez was denied his hat-trick by Theophile-Catherine's goal-line clearance in the closing 20 minutes as Cardiff's hopes faded.
And, despite a much-improved second-half display from the hosts, they are left looking nervously over their shoulders heading into Saturday's match with fellow strugglers Sunderland, giving Tan plenty to think about.