Gibson says that Roberto Martinez can bring Champions League football to Goodison Park.
He said: "Inside the club, there is the idea that we needed a change.
"Moyes wasn't signing his contract for nearly a year, so in that sense the players were expecting him to leave. And given how the fans were upset that he wasn't signing a contract, it was a good thing in the end that he went.
"No one expected him to get the United job but the fact they went for him, when Jose Mourinho was available, says an awful lot about David.
"I enjoyed working under David. He has a sense of humour and got the best out of me.
"But the new manager knows what he is talking about as well.
"Technically and tactically he knows what he's talking about, is very well educated, never seems to be stressed out and is always smiling."