ESPN post-match analysis
Everton edged out the holders in a shootout after 120 minutes failed to separate the sides in the fourth round replay, Leighton Baines having earlier cancelled out Frank Lampard's strike with a dramatic free kick.
Moyes said he was on the verge of replicating former Chelsea manager Mourinho, who is not afraid of leaving his technical area under such circumstances.
"I was thinking about one of those Mourinhos you do down the touchline,'' he said. "Probably making a fool of myself!''
The shootout was not without controversy, with John Heitinga deliberatly bumping into Ashley Cole as the latter walked forward to take a penalty he ballooned over the crossbar, giving Neville the chance to win it.
Moyes said: "Goalkeepers have bits of psychology all the way through, so who knows? Maybe that's the way?''
Asked if there had been harsh words after the Bolton defeat, he said: "Not harsh words, just corrective words, and really realligning what we have to do here before we try to play. Noticably, there was a massive improvement in their attitude to do the tough things first.''
Moyes admitted Saturday's win would "never make up for'' losing to Chelsea in the 2009 final but hopes the omens are good for another run to Wembley.
"I don't know,'' he said. "I've lost to some small clubs and the gulf was even bigger. So how could I say that? I hope you're right but I couldn't be sure.''