Lukaku, on loan from Chelsea, marked his first appearance at Goodison Park with a brace, while England international Barkley was also on target to maintain Everton's unbeaten start in the Premier League.
Belgian striker Lukaku had capped his Everton debut by scoring the winner at West Ham earlier this month and Martinez described the 20-year-old as 'unplayable'.
"What you see today, that's him," Martinez told Sky Sports. "He's strong, powerful, very knowledgeable for a No.9 to have that awareness around him, and he's technically really gifted, so that always brings a very strong package.
"At times he was unplayable. He opens up the spaces for others as well, and we will get stronger. We need to get to know him and the players around him need to get to know him.
"I thought second half at West Ham he showed what he could do and today he looks a player who is enjoying his football and is feeling at home, and that's very important for the future."
Barkley has enjoyed an eye-catching start to the season and Martinez has been impressed by the impact of the teenage midfielder.
The Everton boss added: "It's hard to explain how much he is growing game by game. In pre-season, straight away I was impressed.
"From that point on, he has been getting better and better and better. He is a good listener and is a quite unique talent for an English boy.
"He controls many aspects of the game and normally you don't see that in a youngster like him. He has awareness of space, he has great quality in both feet. He has developed a good understanding with Romelu straight away and that becomes a good focal point for our attacking play."
The win lifted Everton up to fourth in the Premier League table and Martinez is pleased with the start he has made to his reign on Merseyside.
He said: "It has been a good start to the season. I was disappointed with the first three games as we were really dominant and should have got more points.
"It's a great target to get in the top four now, but it's only six games. It gives you a good start, that's all it is. You could see the potential today and some signs are terrific."