The home side opened the scoring through a Virgil van Dijk header in the second minute only for Dele Alli to nab the equaliser 17 minutes later.
It ended 4-1 and Kane was asked on Sky Sports after the match what statement the win has sent out.
Eriksen smashed the bar with a thunderous effort which had Forster stuck to the ground on 71 minutes before Son Heung-min, who had replaced Sissoko, made it 3-1 on 85 minutes.
Kane missed a penalty during the fixture at St Mary's, but the England global netted in the 52nd minute as he returned to the scoresheet alongside Dele Alli, who grabbed a brace, and Son Hueng-min.
Dembele slid Moussa Sissoko in down the left and as the Frenchman's cross span up off Redmond, Alli reacted fastest, his glancing header nestling in the top right-hand corner. Eriksen's long through-pass found substitute Son who took a touch before drilling the ball low into the corner.
Tottenham finished the half strongly and Wanyama danced through the Southampton defence only to be denied by a Jose Fonte block before the break.
- Harry Kane ended a run of 344 minutes without a goal in the Premier League with his strike in this match.
The current Saints chief is preparing his players for a tough festive clash against a Tottenham side sitting in fifth position in the Premier League, but chasing a first away win in the league since September.
Referee Mike Dean did not award a foul against Vertongton, nor did he issue a card, which suggests he may not have seen the incident.
Dele Alli: "To go 1-0 down and come back shows great character".
Alli netted 10 goals in 33 Premier League matches last term, and he claimed nine assists over the course of the season. Southampton remained eighth on 24 points.
Southampton supporters wore orange in support of Koeman when Pochettino first returned to St Mary's in 2015 but the Spurs boss says he takes that as a compliment.