Yaya Toure set City on their way with a third-minute penalty at the Etihad Stadium before Rickie Lambert equalized with a spot kick of his own in the 37th minute.
Samir Nasri and Edin Dzeko both scored in first-half injury time for City. The hosts added an 81st-minute fourth goal through Stevan Jovetic to gain three points that put the team a point behind leaders Liverpool, with City having played a game fewer.
Liverpool visit West Ham today and then host City in a crucial game in the title race next weekend. City can clinch a second league title in three years by winning their remaining seven matches.
Rodriguez’s chances of playing in the World Cup look to be in jeopardy after he was carried off on a stretcher with a knee injury midway through the first half, having landed badly as he jumped for the ball. He is one of the in-form strikers in the Premier League and was likely to be in England manager Roy Hodgson’s plans for Brazil.
“We need to analyse how he’s doing,” Southampton manager Mauricio Pochettino said, “but it’s not looking good.”
Southampton were second best after the break but gave City plenty to think about by levelling at 1-1 with Lambert’s 34th consecutive successful penalty for the club.
The turning point of the match arguably was City’s second goal, which came when David Silva picked up a pass when he was in an offside position before crossing for Nasri to apply a simple finish.
“The linesman killed off the game, it’s clear,” Pochettino said.
“It completely changed the entire game – after that there’s nothing to analyse. We were superior to City in the first half but it’s impossible to fight a decision like that.”
Toure had scored City’s opener after Dzeko was adjudged to have been fouled by Jose Fonte, and Dzeko almost doubled their lead when he headed against the bar from Nasri’s cross.
Southampton’s penalty came when Jack Cork was carelessly tripped by Pablo Zabaleta as he burst into the box and the visitors continued to impress until City’s two-goal counter-punch in added time at the end of the first half. Jovetic came on for Silva to add the fourth, tapping home from a cross by Jesus Navas, and there was still time for Toure to strike a shot against the post from the edge of the box.
“We knew it was going to be a big game and a very difficult one, so we take a lot of positives and confidence out of this game,” City captain Vincent Kompany said. “Not many teams will dominate a game against Southampton.”
City boss Manuel Pellegrini believed his team were worthy winners, irrespective of the controversy surrounding his team’s second goal. He said: “The referee decides, if he made a mistake maybe it was important. But I don’t think the difference today was just one goal.
“They [Southampton] played very well in the first half, it was difficult for us to hold possession.
“But we scored three goals and had another two clear chances to score. I don’t remember they had clear chances to score. I don’t remember Joe Hart making a really important save.
“In the second half we played perfect and had five clear chances.
“Now we have an important game against Liverpool, who have to play tomorrow. We’ll see what happens in the week, but we have a whole week to prepare and after that game we have six more and Liverpool four.”