Rodriguez came into Saturday's contest having scored five goals in his previous four outings to boost his hopes of making England's FIFA World Cup squad, but a nasty-looking knee injury - sustained midway through the first half at the Etihad Stadium - looks set to keep him out of the tournament.
Yaya Toure gave the hosts the lead after just three minutes from the penalty spot after Jose Fonte's lazy trailing leg felled Edin Dzeko in the area.
A similar incident involving Jack Cork and Pablo Zabaleta allowed Rickie Lambert to equalise from the spot, but City regained their advantagelate in the first half as Samir Nasri stroked home David Silva's exceptional pass.
And Manuel Pellegrini's side enjoyed a two-goal cushion at the break through Dzeko - making his 150th City appearance and hailed as a "transformed" player under the Chilean by director of football Txiki Begiristain in the week.
Second-half substitute Stevan Jovetic rounded off the result with just his second Premier League goal 10 minutes from time.
The win keeps pressure on Premier League title rivals Chelsea and Liverpool, with Pellegrini taking his side to Anfield next week for a clash that could decide the destination of the trophy.
City made a barnstorming start, though they had Fonte's inexplicable trip on Dzeko to thank as Toure opened the scoring from the spot, sending Paulo Gazzaniga - playing in place of the injured Artur Boruc - the wrong way.
Nasri wasted a chance to double the advantage two minutes later, and the visitors eventually settled into the game.
Their momentum was checked when Rodriguez was forced off looking in distress after appearing to twist his knee in an awkward landing as England boss Roy Hodgson looked on concerned from the stands.
Despite that setback, Zabaleta then brought down Cork to give Lambert the chance to equalise from 12 yards - and the striker coolly maintained his 100 per cent record from the spot for the Saints.
Southampton had most of the play after their equaliser but were caught with a sucker-punch in the 45th minute as Nasri turned home from Silva's pass, though replays showed the Spaniard had been comfortably offside in the build-up.
Gazzaniga was arguably at fault for City's third, flapping at an Aleksandar Kolarov cross to gift Dzeko his 21st goal of the season.
The hosts controlled much of the second half and had several chances to extend their lead further in the period's opening 20 minutes.
Toure saw shouts for the game's third penalty waved away, Dzeko skewed horribly wide after seeing his initial effort saved and Gazzaniga displayed a better side of his game to keep Nasri out.
The fourth finally came late on as Jesus Navas gave Jovetic a tap-in as City took another step closer to the title, before Toure struck the post late on.
A fourth defeat in Southampton's last five away trips, meanwhile, hinders their late run for the top six.
