The high-flying Merseysiders, who were just two points off the top of the Barclays Premier League at Christmas, slipped to a surprise 1-0 home defeat to the bottom side on Boxing Day.
Yet it was a game that hinged on one critical moment as goalkeeper Tim Howard conceded a penalty and was sent off in the 25th minute. Ki Sung-yeung struck from the spot and Everton, despite creating numerous chances, could not force an equaliser.
Martinez was pleased with the response of his players and feels they learnt more in such circumstances than they would have done in a comfortable win.
The Spaniard said: "In football you don't want to lose games but we can take a lot from the way we faced adversity.
"We lost a man early on and we had to play over 60 minutes with 10 men and I thought the way we did that was magnificent. I couldn't be happier with the performance.
"If you are going to lose a game the only thing you can do is learn from the situation and make sure we are stronger for the next game.
"That is the way we are approaching it and I truly believe we are a stronger side now than we were before the game started.
"You can get many positives from a defeat, in the same way you can get many negatives in a victory."
The Sunderland result denied Everton the chance to complete the calendar year 2013 without losing a home league game.
But Sunday's visit of Southampton marks the halfway point of the season and there is plenty of reason for optimism looking ahead.
Martinez said: "This is always a significant moment to assess where you are and make sure we use the January window well.
"It is a very important moment of the season but we can only be encouraged and excited with the first half of the season.
"The players have shown their potential and we can't wait to carry on developing that until the end of the season.
"We are fortunate with the fans we have, recognising and applauding the efforts of the players. That's what gives you extra points over the course of the season and it couldn't give me more excitement looking forward to the second half of the season."
Southampton have enjoyed a fine start themselves and head to Goodison Park on the back of a 3-0 win at Cardiff.
Martinez said: "Southampton are a terrific side, I think one of the best sides in this competition.
"We know we are going to have to be at our best but we want to get rid of the feeling we were left with after the previous game.
"It is a great opportunity to be on the pitch again.
"We don't want to find an excuse - the opposite. We always look at ways we can overcome the difficulties and perform in a manner that can get us a positive result."
Goalkeeper Joel Robles, who won the FA Cup under Martinez at Wigan last season and then followed him to Everton, will feature with Howard now suspended.
Midfielder Gareth Barry, who has proved hugely influential since arriving on a season-long loan from Manchester City, also misses the game after accumulating five bookings.
Martinez said: "Tim Howard has been sensational the whole campaign and has set high standards on a daily basis.
"Joel is a young man who has been very professional waiting to help the team, and now he is going to have an opportunity to do that.
"The whole squad, since day one, has been working with the same mentality of being ready and supportive of each other.
"We couldn't be happier with the goalkeepers at the club.
"In the same way we are missing Gareth Barry, another experienced footballer who has set high standards, but it becomes an opportunity for someone else.
"We'll never use an excuse not to perform. It is an opportunity for the players who haven't been playing to show how well they have been working behind the scenes."
England left-back Leighton Baines is in contention to return after more than a month out with a broken toe.