The visitors performed well for large spells but were undone by two good finishes and found it difficult following Elliot Ward's dismissal.
The winger found City's top scorer just inside the box, and he made no mistake, curling the ball past the goalkeeper and into the back of the net.
Nugent then had a header cleared off the line before a stunning volley from Marc Pugh brought Bournemouth level as the winger came in from the right and thumped the ball into the top corner.
Anthony Knockaert almost put the home side ahead just after half time with a well-struck drive, which was tipped away.
They didn't have to wait long for another chance though, and this time they took it, as Jamie Vardy collected a through ball and rounded the goalkeeper before slotting into an empty net.
Bournemouth's day got even worse moments later when Ward was shown his second yellow card for a cynical trip on Chris Wood.
Leicester were comfortable thereafter and saw out the game easily to take all three points.