Mark around the damaged area with a square and pencil. Make the mark square, with two of the opposing sides running along ceiling joists. You can determine the joist positions by pulling down some of the plaster and looking inside the ceiling.
Cut around your marked line with a jigsaw. The cuts that are on the joists should expose half the joist widths.
Measure the dimensions of the opening and transfer them to a piece of drywall, making them about 1/8th inch smaller all around then the actual dimensions. Cut out the drywall by scoring it at the lines with a razor knife and snapping it off.
Lift the cut out drywall up to the hole, with two the two opposing sides of the square sitting on the half-exposed joists. Secure it by driving screws in every six inches or so along the edges that have joists behind them.
Lay strips of adhesive mesh drywall tape along each seam between the drywall square and the surrounding ceiling. Use a wide drywall knife to spread your joint compound over the seam, getting it smooth and flat.
Let the compound dry overnight. Sand it smooth. Apply a second coat of compound, let it dry and sand it. Repeat a third time. Repaint the ceiling.