The Rhinos went in front after 10 minutes thanks to Stevie Ward, who touched the ball down beyond the try line after a well-timed run before Kevin Sinfield converted.
Into the second half, and the Giants went ahead when a high kick dropped to Cudjoe who found Jamie Ellis wide open to run the ball over.
Danny Brough converted, and secured a further two points with a penalty shortly after to make it 16-8 to the home side.
Leeds got a second try back with time running out as Tom Briscoe slid over and Sinfield converted to make it 16-14 with just over five minutes remaining.
With 53 seconds until the final whistle was due, Sinfield kicked a drop goal to level things at 16-16.
Video referees were then required to confirm that Ryan Hall had won the game for Leeds with a try in the final second to secure the League Leaders' Shield and round off an exhilarating end to the game.