Chelsea remain unbeaten in the seven matches since new boss Thomas Tuchel replaced Frank Lampard but failed to open up a gap in the top four race.
"We were very, very good in the first 80 metres of the pitch but in the last 20 I was not happy," Tuchel told BT Sport. "We were not decisive enough, we were not aggressive enough."
Chelsea started the game well and almost opened the scoring after six minutes but Marcos Alonso's close-range volley went wide.
The visitors dominated possession in the early stages but failed to create many clear-cut chances.
Chelsea's lack of cutting edge was punished after 33 minutes after Nathan Redmond played through Minamino who skipped past Cesar Azpilicueta and Edouard Mendy to put Southampton ahead against the run of play.
Chelsea were awarded a penalty shortly after the break when Mount was brought down by Danny Ings and the 22-year-old was on hand to convert the spot kick.
Southampton almost went back in front after 70 minutes but Jannik Vestergaard's looping header hit the crossbar.
Chelsea had a great chance to score a late winner but Reece James blazed over from Mount's cross.
Tuchel defended his decision to bring off Hudson-Odoi and said he was "not happy" with the forward's cameo performance.
Southampton boss Ralph Hasenhuttl was proud of his team and said: "With a little more luck in the end we can nearly win the game with the header of Jannik. But I think the draw is for us like a win today."