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Radzinski was the man behind both the goals. Unsworth's came from a penalty in the 56th minute after Danny Mills realised he couldn't match Radzinski for pace and decided to push him to the ground.
Unsworth calmly stepped up to the penalty spot and slammed home to the right of the keeper - setting Everton on their way to a well deserved victory.
Then in the 68th minute Chinese international Li Tie combined with the Canadian to create space in the box.
And with plenty of time Radzinski controlled the ball, looked up and fired to the right of Paul Robinson.
It was a different story in the first 45 minutes as both sides had efforts at goal but they were few clear chances.
Neither team dominated and the only genuine effort fell to Everton's goal-scoring midfielder Steve Watson.
With just six minutes on the clock the 28-year-old latched on to a neat centre by Brian McBride but he failed to add to his tally of four goals in four starts blasting his shot over the bar.
Following the controversial sales of Robbie Fowler and Jonathan Woodgate the Leeds faithful are baying for the board's blood and the repertoire of chants calling for Peter Ridsdale's head more than made up for the lack of imagination shown by the players on the pitch.
The were glimpses of skill for Alan Smith, Harry Kewell and Jason Wilcox but they couldn't fashion anything coming close to a goal.
In the second half it was all Everton. The visitors were as short of ideas as they are cash and Toffees goalkeeper Richard Wright had an easy day at the office.