Goals from Kevin Mirallas after six - from a slick passing move - and 60 minutes wrapped up a comfortable victory for the Toffees.
The usually stoic Moyes, set to replace Sir Alex Ferguson at Old Trafford in the summer, was visibly moved by the reception he received both before, during and after his final appearance at the ground he has called home for 11 years.
Allardyce said: "The way we played we were lucky to get away with 2-0.
"There was only one player who has been outstanding for us and that was (goalkeeper) Jussi Jaaskelainen, who made some outstanding saves.
"I warned the players before the game what it was going to be like - it was exactly that and the players couldn't cope."