Phil Jagielka prodded the ball home from close range early on at Goodison Park on Saturday, and although the visitors subsequently dominated possession, Everton defended stoically to claim the spoils.
Martinez, who has endured a difficult second season at the helm on Merseyside, has called on his team to continue their momentum.
"We want to maintain this winning focus and intensity," he said. "We know how important the wins are at this stage of the season and today we carried it [their momentum] on with the character and the grit we have shown in the last three matches."
"It was a tactical battle where both teams had to adapt their style, and from our point of view a really open game.
"Early on, they had two really good chances, but [goalkeeper] Tim Howard was at his best and then we defended really well and Southampton didn't have another shot on target.
"Overall, after we scored the goal, I think we managed the game excellently. We showed a real maturity in our play and what we have learned in this tough season."
Howard has come under heavy criticism from sections of the Goodison Park faithful this term, with the former Manchester United shot-stopper having made several costly errors.
But he was back to his best to deny Graziano Pelle from scoring his first league goal of 2015, the American back-pedalling superbly to claw a looping header off the line with one of the saves of the season.
Jagielka put the hosts ahead soon afterwards, tapping in from close range following good work from Gareth Barry in a crowded penalty area.
And Martinez was quick to single out the contribution of his match-winner at both ends of the pitch.
"Phil Jagielka has performed at a high level for us and especially in the last three games against Newcastle and QPR," added the Everton boss.
"As a captain, that's one of the best displays I've seen him put in for us, and the whole team responded really well."
