Mikel Arteta's 60th minute spot-kick, given after Jean Beausejour was adjudged to have felled Walcott, proved the decisive moment in a very tight game played in sodden conditions.
Martinez said: "When I saw it live it looked very, very soft because it's difficult to tell if there is contact or not, and that doesn't mean it's a penalty, the contact has to stop you from staying on your feet."
He added: "The player's got real experience and he knew a penalty would change the game and give them real hope of the three points and I thought he bought the decision.
Referee Jon Moss made himself even more unpopular with the hosts when he turned down a handball shout in injury time. Substitute Jordi Gomez's shot certainly struck Kieran Gibbs' arm, and Martinez felt it was a clear case of his side being on the wrong end of two decisions.
"Whether it's a soft penalty or not, the disappointment is then in the other box you've got a clear handball," Martinez said. "The ball travels a long distance and the arms are not in a natural position, and that's a penalty. It's disappointing you don't see that consistency. But I think it would be very unfair just to talk about those two incidents."
Arsenal manager Arsene Wenger did not see that incident but was pleased his players showed they could grind out a victory on the back of Monday's thumping 5-2 win at Reading.
He said: "They organised to stop us playing and they did that very well, especially in the middle of the park. We didn't find the same fluency going forward and it was much more a win because we wanted the result than because we dominated the game."
Walcott was again given his chance up front after impressing in the role against Reading, and Wenger gave the England man qualified praise.
He said: "I wouldn't like to judge him on one game but he has the ingredients to play there. He didn't have much service today, he was a bit isolated but he can always be decisive. Today again he made the decision and he got the penalty."
Source: PA
Source: PA