Everton lost ground on the pacesetters in the Premier League after suffering a 1-0 loss against struggling Sunderland at Goodison Park.
Goalkeeper Tim Howard was sent-off moments before Ki Sung-yeung struck home the winning penalty, but Everton produced a battling response and were still pushing for a leveller in the closing stages.
Martinez remained upbeat after the defeat and feels his players have proved they can dig deep in tough circumstances.
"In football you don't want to lose games but we can take a lot from the way we faced adversity," said the Spaniard.
"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."
Southampton will head to Merseyside with plenty of confidence after their 3-0 win over Cardiff on the road and Martinez is full of respect for Mauricio Pochettino's men.
"Southampton are a terrific side, I think one of the best sides in this competition," he said ahead of Sunday's clash.
"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."