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 of the penalty: "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.
"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.
"But whether it's a soft penalty or not, the disappointment is then in the other box you've got a clear handball. 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.
"When you play against Arsenal and they're running the clock down in the second half and making substitutions to try to cope with the threat we had, it's a real compliment.
"You're not going to see many teams having more possession than Arsenal and I think my players deserve an incredible amount of credit.
"We didn't get any luck and sometimes in football it's better to be lucky than good but I'm very proud and if we can carry on with this level over the next few games we'll get the rewards we deserve."