Goals from Seamus Coleman and Romelu Lukaku either side of a 30-yard piledriver from Gaston Ramirez secured all three points as the Toffees recovered from their Boxing Day humbling at the hands of Sunderland to end the year on a high note.
Martinez declared himself very pleased with his side's season so far but refused to set any specific targets, telling reporters: "I think we can improve.
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"It is impossible to set a target in December. All we are doing is taking it game by game, trying to get as many points as we can going into the final third of the competition."
Martinez made several changes to his starting line-up as a result of suspensions, injuries and squad rotation and was delighted with the way in which his altered side coped.
"You're not going to see many teams making those sort of changes and being able to perform with the unity and understanding that we had," he enthused.
"From now until the end of the season we have got the majority of the top sides coming to Goodison and that is a real challenge to be able to fulfil our potential - I don't think we have done that yet."
Meanwhile, Southampton boss Mauricio Pochettino praised his players for the way in which they took the game to Everton for long periods.
"I think, overall, the team played very well," argued the Argentine. "There were many phases of the game where we were actually superior to Everton.
"It is clear that we are a young team that has personality and an attacking style and always goes for the win.
"We all saw what happened; we deserved a lot more from the game. From the three shots that they had on goal, they scored twice. We created a lot of chances."