Lukaku, who is on loan at Everton from Chelsea for the season, struck the winner in the 74th minute of the Barclays Premier League match at Goodison Park when he converted from James McCarthy's pass.
The goal, the Belgium striker's ninth for the Merseysiders this term, was his first in six games, but Martinez disagreed with the suggestion that Lukaku "needed" to get back on the scoresheet, 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."
It was full-back Seamus Coleman who put Everton ahead in the ninth minute, racing into the box from the right and unleashing a powerful shot into the top corner of the net.
Southampton substitute Gaston Ramirez equalised in the 71st minute, when he beat keeper Joel Robles with a speculative shot from distance. But three minutes later, Lukaku struck to secure the three points.
At St James' Park, Arsene Wenger praised Arsenal's new-found resilience as they withstood a late onslaught by Newcastle to climb back to the top of the table. The Gunners had to defend deep and in numbers as they clung on to the advantage given to them by Olivier Giroud's 65th-minute header, taking three precious points in the title race.
Wenger said: "There is something in the team that is special on the mental front. We have many times been questioned, but we have shown that at West Ham, where we were 1-0 down, we have shown that again today, where we were a bit backs-to-the-wall in the last 15 minutes. There's a great solidarity and a great spirit in the side."
At White Hart Lane, manager Tim Sherwood tasted victory for the first time in his brief spell in charge as Tottenham Hotspur put on a vintage display to crush Stoke City. Roberto Soldado converted his fifth penalty of the campaign while Aaron Lennon and Mousa Dembele also scored to give Spurs a comfortable 3-0 win.
Tottenham were flat and wasteful in the 1-1 draw against West Bromwich Albion on Boxing Day, but yesterday Dembele and the returning Paulinho bossed the midfield against a physical Stoke side, while Lennon gave Tottenham the cutting edge they lacked against West Brom.
Sherwood promised to bring attacking football back to the blue-and-white half of north London and he certainly has lived up to that vow. His team have averaged two goals a game in the four matches he has been head man in the Tottenham dug-out.
But more welcome to the Tottenham manager will be the fact that his team managed to keep a clean sheet in the league for the first time since November 3.