The Merseyside outfit had not beaten Manchester United in their own backyard for over 21 years before Bryan Oviedo struck four minutes from time in Wednesday's Premier League clash to stun Moyes.
The Scotsman spent 11 years at Goodison Park before being charged with the task of replacing Alex Ferguson and Martinez achieved what his predecessor failed to in Moyes' first game against his former employers.
Martinez was delighted with the manner in which his side took the game to United and expressed his pride after they stretched their unbeaten run to seven top-flight games.
He told Sky Sports News: "I have immense pride, because it wasn't just a game of football. It was a mental block that we had to overcome, because over the years we haven't been able to come to places like this and win.
"To be able to come here and play with the arrogance we have throughout the game. I'm really pleased and satisfied with the hard work that we put in."
Costa Rica defender Oviedo has now scored two goals in as many games since replacing the injured Leighton Baines.
And Spaniard Martinez is delighted to be able to call upon such an impressive replacement in the absence of the injured England left-back.
He added: "It tells you everything about Bryan because he's got the hardest job, to be understudy to one of best defenders in world football."
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.