Timo Werner and Mason Mount overcame recent struggles to star in the comprehensive win, both netting braces along with goals from Marcos Alonso and Kai Havertz.
Alarm bells had begun ringing in west London after a worrying 4-1 league loss against Brentford was followed by a 3-1 defeat to Real Madrid in the Champions League quarter-finals.
Southampton though, provided no threat at all, rolling over as Chelsea assailed to a 4-0 half-time lead.
Mount showed two moments of class to first assist Alonso for the opener, before adding his own long-range effort soon after.
Werner then scored what may be the best goal of his career in England as he burst down the left channel with scintillating pace before leaving Fraser Forster in a heap.
It took Chelsea just 31 minutes to go four up, the quickest any side has taken to race into such a lead in the Premier League since the Blues last did it 2011.
Werner and Mount then both doubled their tallies in the second-half to give manager Thomas Tuchel plenty to build upon before his side travel to Madrid in a season-defining clash.
A Karim Benzema hat-trick in the first leg of their European quarter-final massively dampened the mood at Stamford Bridge, but Chelsea's away form gives plenty of reason for optimism.
The Blues have now won their last seven on the road in all competitions, the joint longest streak in the club's history, boosting confidence before they head to the Spanish captial.
As for Southampton, there was little reason to be positive after Hasenhuttl's tenure was marred by another trouncing.
The Austrian owns two of the three worst losing margins in Premier League history, falling 9-0 to both Leicester and Manchester United during his tenure.
A second 45 where Chelsea took their foot off the gas helped the Saints avoid another record defeat, but ensures there will be plenty of question marks around their coach as the season comes to a close.