Lukaku, who is on loan at Everton from Chelsea for the season, struck home what proved to be the winner in the 74th minute of the Premier League clash at Goodison Park when he converted from James McCarthy’s pass.
The goal, the Belgium striker’s ninth this term, was his first in six games.
But Martinez disagreed after the match with the suggestion that Lukaku “needed” to get back on the scoresheet, with the Spaniard being keen to highlight other aspects of the frontman’s game.
“From the outside, yes, you are going to judge a striker by his goalscoring ability and the numbers,” Martinez said.
“But for me, if he scores goals, great — but it is a bonus. I judge him in terms of all the aspects. I thought he was magnificent against Swansea [a recent 2-1 win] without scoring.
“I couldn’t be happier with his finish today, but I think that is an add-on.”
Asked if Lukaku had been getting down at all about his run of matches without scoring, Martinez added: “No we have been working on other aspects of his game.
“We have never been in a position where we were getting a little bit worried and thought that was his only role as a striker.
“You want strikers to score goals, but in the same way, he attracts two or three opposition players at the moment, so it is understandable that other players get that stage – players like Seamus Coleman, who gets into that position really well and takes advantage of it.
“It is a team effort and Romelu has not been down at all – it has been the opposite.”
It was full-back Coleman who put Everton ahead in the ninth minute. The Republic of Ireland international collected the ball from Leon Osman, advanced down the right flank and then cut into the box, beating James Ward-Prowse in the process, before unleashing an emphatic effort beyond Kelvin Davis and into the net.
Southampton substitute Gaston Ramirez equalised in the 71st minute when he beat Toffees goalkeeper Joel Robles, who it appeared should have done better in his attempt to save the effort, with a shot from distance.
The three points moved Everton up a place to fourth on 37 points, a point and a place above Liverpool who later lost at Chelsea.
And asked for his thoughts on how his team were faring having completed their 19th fixture of the 38-game league campaign, Martinez said: “37 points is an outstanding return half-way through the season.
“But I think there is room for improvement and that is the target we have from now until the end of the season.
“I don’t think we have fulfilled our potential yet. You will always hear me saying it is impossible to set a target in December. We are just trying to get as many points as we can.”
Southampton, who remain ninth in the table, certainly had their moments in the match and their boss Mauricio Pochettino felt his side had been unfortunate to end up empty-handed.
“I thought overall it was a very good game, with both teams going forward and playing good, attacking football,” Pochettino said.
“But it is a shame, because we leave with nothing and I think we deserved a lot more from this game.
“My team played very well. There were many phases of the game in which we were actually superior to Everton.
“It is clear that we are a young team, but this team has personality and a certain attacking style, and we try to impose this wherever we are playing, always going for the win.”
Meanwhile, Everton captain Phil Jagielka will be out for up to four weeks due to a hamstring problem.
Jagielka missed yesterday’s victory, with the Merseyside club stating on their official website beforehand that a hamstring injury had been the reason.
And after the match, when asked about the England defender, Martinez said: “It was at the end of the game against Sunderland (Everton’s previous fixture, a 1-0 home defeat on St Stephen’s Day).
“With this soft tissue injury it could be up to four weeks, but knowing Jags he will be back earlier. It is just the accumulation of games and the demands of this month — November into December is high-risk in terms of injuries to the players. When you have the number of games we play, it is understandable.”
EVERTON: Robles, Coleman, Alcaraz, Distin, Baines, McCarthy, Osman (Mirallas 72), Naismith, Barkley, Oviedo (Pienaar 60), Lukaku.
SOUTHAMPTON: Kelvin Davis, Chambers (Clyne 88), Fonte, Lovren, Shaw, Ward-Prowse (Gallagher 84), Cork (Ramirez 66), Rodriguez, Steven Davis, Lallana, Lambert.
Referee: Mark Clattenburg.