AFC Bournemouth's 19-year-old debutant goalkeeper Mark Travers denied Lucas Moura, Son Heung-min and Dele Alli in the early stages.
On the stroke of half-time, Spurs were reduced to 10 men after Son shoved Jefferson Lerma.
Only two minutes and 13 seconds after coming on as a half-time substitute, Juan Foyth was also sent off for a challenge on Jack Simpson.
After substitute Lys Mousset twice went close, Ake headed in Ryan Fraser's stoppage-time corner to secure the Cherries' first ever PL win over Spurs.
AFC Bournemouth move up to 13th on 45 points.
AFC Bournemouth: 12 May v Palace (A)
Spurs: 12 May v Everton (H)