Baines struck two brilliant second-half free-kicks to twice draw his side level, before Romelu Lukaku popped up to head home a winner, and secure Everton's status as the only unbeaten team in the Premier League.
And Martinez admitted that he was not surprised by the England left-back's performance.
He said: "I think we are all well aware of the quality that he (Baines) has and the first free-kick would show you that.
"I think that where Leighton takes his quality level into a next step is that he's capable of taking another free-kick with a world-class goalkeeper with the experience that Jussi Jaaskelainen has and score.
"There are many footballers around the world that have a really good quality to take those free-kicks but when you put Leighton Baines into that situation the second time, and to be able to do that I think it's quite unique."
Despite the win, Martinez admitted that his side were too slow out of the blocks in the first half, which saw them fall behind to a Ravel Morrison goal.
He added: "We made it hard for ourselves, but I think you have to credit West Ham, they wanted to stop us from playing.
"It was a little bit forceful our play, and the first time we were able to get through them we created a really good chance that we should have scored.
"I think we weren't as sharp as we were on Saturday (against Chelsea). That performance took a lot from certain players and I think being refreshed in the second half really helped us.
"We defended really well. We stopped West Ham being a threat. I don't think we were as good as we can be on the ball but clearly with the amount of chances we created in the second half and the quality that Leighton brought in dead-ball situations...
"It was a well-deserved win in the end."
