Mark a square around the damaged plaster. Mark two sides along ceiling joists on either side of the area. (You can find the joists by pulling down enough of the damaged plaster to look up into the ceiling.)
Cut out the square with a jigsaw, sawing through both plaster and lathe. Cut two sides of the square along the joists, exposing half of the width of the joists, leaving plenty of joist to attach the drywall to.
Measure the thickness of the combined plaster and lathe at the edge of the hole--this will tell you what thickness of drywall you need. Measure the hole in the ceiling and mark the dimensions on a piece of drywall. Cut the drywall with a utility knife, scoring along the lines, then snapping the piece.
Set the piece of cut drywall in place. Sink drywall screws through two sides and into the exposed joists, spacing the screws every six inches.
Press mesh drywall tape over all four seams, between the drywall and the surrounding ceiling. Spread plaster over the tape with a drywall knife, getting it smooth and flat. Let the plaster dry for six hours.
Sand the plaster smooth with a pad sander. Spread a thin, flat layer of plaster on and let it dry. Sand the second layer and add a third. Sand it smooth and paint the ceiling.