The Saints sacked Mauricio Pellegrino on Monday after one win in 17 games left them a point and a place above the Premier League relegation zone.
Southampton have won just one of their last 17 league matches - against bottom club West Brom - with Saturday's 3-0 defeat at Newcastle proving to be the final straw.
Assistant manager Carlos Compagnucci and assistant first-team coach Xavier Tamarit have also parted company with the south coast club.
"We would like to place on record our thanks to Mauricio, Carlos and Xavier for their efforts during their time with Southampton, and wish them well for the future", the club said in a brief statement.
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Although the Liverpool game was Rashford's first league start this year, he has played 39 games for United this season has scored 12 goals.
The sacking of Pellegrino raises questions once more about Southampton's decision to fire Puel last summer.
Now, it's claimed that Southampton are turning their attention to Celtic boss Rodgers, and may hope to lure him back to the Premier League after almost two years in Scotland.
Saints will be hoping to appoint Pellegrino's successor quickly, with an FA Cup quarter-final at Wigan looming on Sunday. The team has been in the Premier League since 2012.
"I think they may have got a little but too complacent and thought they are too good to be in this position go down", the Saints legend added.
Former Watford boss Marco Silva was instantly linked with a swift return to the top flight, but he is no longer the favourite to take the job.