The Barcelona striker won the penalty after being brought down in the box just 13 minutes into the second-half.
Lewandowski, who has already scored 18 goals in 19 games for his club side this season, stepped up to take the spot kick but failed to convert as Mexico goalkeeper Guillermo Ochoa pulled off an excellent save.
The Barcelona ace revealed he changed his penalty-taking technique and was 'hurt' by the save. Speaking after the match, he told TVP: "It was an even game, both teams could have won.
"On the one hand, we will respect this point. On the other - there was the penalty and I didn't score it. It hurts. In the next match we have to win. We all know this."
He added: "Did I change my technique for the penalty? Yes. Why? That was my decision to kick the ball to the side. That's how it is with penalties: if everyone knew how to kick the ball before taking the shot, it would be easy. Today I failed. It hurts."
The match between Poland and Mexico was arguably the dullest affair of the tournament so far, with a mere five attempts on goal.
Poland created just one of those attempts as Lewandowski struggled for service during the encounter.
He said: "We're working a lot defensively so that the opponent doesn't create dangerous goal situations against us. But we still need to improve on things.
"Sometimes there is a lack of strength, good decision making, or faith that we can score a goal.
"The more our players attack, the easier it is for me to get goal opportunities. If we have in the back of our heads that the defence is a priority, then there are no clear goal chances made. We have to strike a balance between defence and attack.
"There were lots of long balls, many times I won duels for them, but there was a lack of collecting the ball. It was not an easy match for our attacking players. We were pressed, doubled. With a more daring game, we are able to create goal chances."
Poland will now set their sights on history-makers Saudi Arabia who stunned Argentina with a 2-1 win on Tuesday.