Jamie Vardy had slid the ball in from two yards to put the Foxes in front after 23 minutes, with his 23rd goal of the campaign.
But half-time substitute Junior Stanislas made it 1-1 on 66 minutes from the penalty spot, slotting down the middle after Wilfred Ndidi fouled Callum Wilson.
And AFC Bournemouth were in front 101 seconds later after Kasper Schmeichel was unable to keep out Solanke's low shot, with Caglar Soyuncu sent off in the aftermath for kicking Wilson.
Jonny Evans' own goal in the 83rd minute and a second for Solanke four minutes later wrapped up the win.
A first victory in 10 matches leaves the Cherries three points from safety in 18th with three fixtures to go.
AFC Bournemouth: 15 Jul v Man City (A)
Leicester: 16 Jul v Sheff Utd (H)