Goals from Pape Matar Sarr, Heung-min Son and Richarlison ensured Spurs responded to their 4-2 defeat to Brighton with a victory.
Late on in the match, Alejo Veliz picked up an injury, with Postecoglou and Giovani Lo Celso telling the player to go down for treatment.
A member of the Bournemouth coaching team took exception to this and appeared to shout at the Spurs boss in retaliation.
There was a coming together on the touchline and Bournemouth boss Andoni Iraola was seen pushing his fellow coach out the way.
Arguments between the two sides continued and Postecoglou, as well as Lo Celso, received bookings.
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Veliz eventually had to make way and was visibly emotional when being substituted, with Postecoglou consoling him.
The Spurs boss spoke on the touchline clash after the match, and certainly viewed it in a light-hearted way: "We all just came together to wish each a Happy New Year and make some resolutions."
Spurs were able to cap off 2023 with a win, but it was far from a perfect afternoon for them.
Sarr opened the scoring after nine minutes but was substituted through injury just after the half hour mark.
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The midfielder covered his face with his shirt, with his chances of playing for Senegal in the Africa Cup of Nations now in doubt.
Son doubled Spurs' lead in the 71st minute, with the winger playing his last game before he represents South Korea at the Asia Cup.
Richarlison sealed their win not long after before Alex Scott scored a consolation for Bournemouth to complete the scoring.
Spurs' win has taken them to within a point of north London rivals Arsenal, who lost 2-1 against Fulham.