The Toffees fell behind twice at Upton Park before the game turned in their favour in the final nine minutes.
West Ham were left to rue the dismissal of Mark Noble for a second yellow card, with Leighton Baines crashing home his second spectacular goal from the resulting free-kick and Romelu Lukaku bravely nodding in what turned out to be the winner shortly after.
Martinez was delighted to see his summer transfer business pay dividends - with Lukaku drafted in on loan from Chelsea and Baines persuaded to stay on at Goodison Park amid interest from Manchester United - but the Spaniard was also quick to hail the collective effort of his troops.
He told Sky Sports: "As a team, I did feel that West Ham did very well early on trying to stop us from playing.
"But then all of the things you have to do in a game, I think we have limited West Ham to three shots on target and one is a penalty. To do that away from home takes a lot of work rate and I think the squad reflects the effort of the whole football club.
"Keeping the best players is important, being able to bring in the players of the youth and experience that we have is giving you a good opportunity and a perfect blend. But the talent without the mentality and the character gives you nothing, and today we had to show a lot of character."
Asked if managers take as much pleasure from a hard-fought win on the road as they do from a home victory over a side such as Chelsea - which Everton enjoyed last weekend - Martinez said: "You do. You know when you come to Upton Park that it's going to be a really tough game. When you concede two goals, you put yourself in a difficult position.
"In the second half we showed an incredible will and desire and a winning mentality. Probably in the first half we were a little bit pedestrian and we couldn't get fully on the ball - maybe the effort of last week cost us a little bit.
"But the second half, I thought Romelu and James (McCarthy) when they came on really helped us. It was very pleasing to get the points in the manner we did."