Martinez achieved at the first time of asking what his predecessor David Moyes, who is now at the helm of the Red Devils, failed to manage in his 11 trips to the Theatre of Dreams.
Bryan Oviedo scored the only goal of the game in the 86th minute of an absorbing end-to-end contest which saw the woodwork struck three times, with Wayne Rooney, Danny Welbeck and Kevin Mirallas all denied by the frame of the goal.
And with Everton now fifth in the table, just three adrift of Chelsea in second and five clear of United, Martinez feels the only way is up for the Goodison Park club.
Describing his post-match feelings to Sky Sports, he said: "It was immense pride, because it wasn't just a game of football, it was a real mental block that we had to overcome.
"Over the years we haven't been able to come to places like this and get the three points.
"Being able to play with the arrogance that we had throughout the game, I'm really pleased and proud of the players and the performance and really satisfied with the hard work that we put in."
Martinez reserved special praise for match-winner Oviedo, who has stepped into the breach with regular left-back Leighton Baines on the sidelines with a broken toe and grasped his opportunity with both hands.
"I think it tells you everything about Bryan as a professional, he has got one of the hardest jobs, to be the understudy for one of the best left-backs in world football," he added.
"When you get the opportunity, you get a statement of what he has been doing behind the scenes. Bryan has been an incredible professional, biding his time and being ready for the team.
"The two performances couldn't be a better reflection of who he is as a player and a human being - he deserves everything."