Maddison put Spurs in front with his first goal for the club and the midfielder was outstanding at the Vitality Stadium.
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.
Yves Bissouma and Pape Sarr linked up on the edge of the area, Sarr picked out a run through the middle from Maddison and he guided the ball past Bournemouth keeper Neto.
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.