Spurs had to weather a storm at the start of the second half but they withstood some Bournemouth pressure and Dejan Kulusevski then doubled their lead.
Maddison scored after 17 minutes when he finished off a good move.
Maddison was everywhere and he then slid in Richarlison, who should have done better but never had the ball under control and Bournemouth scrambled clear.
Philip Billing had a great chance at the end of the first half but he bent a shot inches wide.
Spurs were on the back foot at the start of the second half but Kulusevski added a second on 63 minutes when he turned home a cross from Destiny Udogie.