Cazorla opened the scoring in the sixth minute but Villa came back strongly and equalised through Andreas Weimann after the break.
Arsenal threw everything at Lambert's men but they could not find a way past Brad Guzan until Cazorla found the net with five minutes remaining.
The defeat saw Villa slip into the bottom three on goal difference following Wigan's win at Reading.
Lambert was disappointed to see his team come away with nothing despite a pleasing display.
"We shouldn't have lost it," the Scot said. "We were exceptional in parts of that game and we should have been in the lead at half-time.
"At the very least we should have come away with a point. We've played like that for a number of weeks: West Brom, Everton, Newcastle in the second half, West Ham.
"We've just got to try and get more points on the board. We've got massive games still to come, and there'll be a lot of twists and turns this season."
Lambert's side have only won once in the Barclays Premier League since December but he is confident they will not be relegated this season.
"I have the belief in ourselves that we'll always be okay," the former Norwich manager said. "I don't get caught up in the whole issue. If the lads keep playing like that, that's all I can ask for."
- Sports & Recreation
- Andreas Weimann