It showed the size of the task facing Ralph Hasenhuettl, the Austrian who watched from the stands after being hired earlier in the day by Southampton to replace the fired Mark Hughes.
He will have been encouraged by his team's performance when Tottenham eased off in the final 30 minutes and the visitors got a late consolation through Charlie Austin.
With Hasenhuettl assessing his players, left back Matt Targett produced some lax defending that allowed Spurs to take a ninth-minute lead.
As the hosts played a corner short, Targett fatally did not match Kane's run to the near post and the striker netted from Christian Eriksen's cross. It was his ninth league goal of the season and seventh in six games against Southampton.
Two goals in the opening 10 minutes of the second half killed the game as Tottenham hit back after losing the north London derby to Arsenal on Sunday.
After Eriksen's low free-kick was well saved by Alex McCarthy, Moura fired home from the resulting corner at the second attempt for his fifth goal of the campaign.
Then Targett was the culprit again as Kieran Trippier dispossessed the defender and passed to Kane, whose low cross was turned in by Son.