After a goalless first half at St Mary’s Stadium, Pep Guardiola saw Kompany head in his first Premier League goal since August 2015 from David Silva’s corner.
City doubled their lead with 13 minutes remaining when they broke and Kevin De Bruyne passed to Sane to score easily.
Less than three minutes later it was 3-0, De Bruyne crossing for Aguero to head in.
The win lifted City above Liverpool, who play West Bromwich Albion on Sunday, into third. Southampton remain ninth.
Man of the match: Vincent Kompany