The Foxes scored twice in three first-half minutes, before Lewis McGugan halved the deficit from the spot in the second half.
Leicester keeper Kasper Schmeichel was also sent off after the penalty had been converted for throwing the ball away.
Forest were awarded a penalty when Matt Mills handballed a Chris Gunter cross and McGugan fired past Schmeichel, who saw red straight after.
Boateng completed Forest's comeback by scrambling in at the far post in the 93rd minute.
Forest and Leicester now sit at 11th and 14th respectively.