Arsenal failed to break down the Black Cats, whose point at the Stadium of light lifted them out of the English Premier League's bottom three.
Petr Cech had to save sharp efforts from Wahbi Khazri and Jermain Defoe, with Arsenal failing to fashion the same clear-cut chances of their opponents.
The result leaves Arsenal fourth, five points clear of Manchester United - who have a game in hand - leaving Wenger fearful of a run-in that features games against fellow European chasers Manchester City and relegation-threatened Norwich City.
When asked if he was worried Arsenal could lose their top-four spot, Wenger said: "Yes, of course. We care about that and we worry about it as well because it's a fight.
"There are two leagues at the moment: the teams who are taking it a little bit easier - you see some games and you think you would like to play them now, the teams who are safe and are not going for Europe, and then you have the teams who are going for something at the front and the teams who are fighting not to go down, and they are different games.
"It is frustrating because we play to win the title and the fact that we do not win it is of course frustrating, like it is for many other teams.
"A team like Sunderland is ready to fight for their lives, and of course they will never give up during the 90 minutes.
"They started with a bit of anxiety and we had to take advantage of that level of anxiety they had at the start of the game, but we couldn't and you know with the fact that you have played Sunday, Thursday, today that you could suffer a little bit physically in the second half and that's what happened.
"Then the game was much more difficult for us."
