Arsenal are now in Thursday's draw while they are guaranteed around £25 million as a reward for reaching the group phase, easing mounting pressure on the club following Saturday's 2-0 Premier League loss against Liverpool.
Wenger said: "We live in a society where everybody has an opinion on everything. I'm like somebody who flies a plane for 30 years and I have to accept that somebody can come into the cockpit and thinks he can fly the plane better than I do.
"But that's part of our job and we have to accept that. I just would like to say that the club is in overall a very strong position because tonight, for 15 years on the trot, we play in the Champions League.
"We have a new stadium and a fantastic training ground, a very good financial situation and a very strong team."
He added to ITV1: "(There is) relief of course, we don't want to go out - and pride as well, because we had our backs to the wall and responded very well in the second half.
"There was only one team in the second half - Wojciech saved the penalty, which was very generous, but we played a great game tonight. It showed that under pressure this team can respond with football, remain calm and composed and try to play like we can. We had a great belief in the way we want to play.
"There are many questions around our club, if we had gone out tonight it would have been much worse. It was important not financially, like many people have said, but more for the fact we want to play at the top level."
Speculation has surrounded Arsenal's ability and intent to compete in the transfer market ahead of next Wednesday's deadline, but Wenger said: "We work very hard, I have said that many times, but we have many players out as well so I think we can be very positive as well.
"There may be signings but I cannot give you any names tonight. I cannot lie and say we sign somebody tomorrow."
Reflecting on his side's exit, Udinese boss Francesco Guidolin said: "First, we need to accept that Arsenal is a stronger team. They came here and showed great personality and I'm happy we kept the result so close over the two matches.
"If we would've scored the penalty, there would've been 20 minutes of fire on the pitch. But after that, we stopped playing.''
• Catch action from every Barclays Premier League match on your mobile for FREE on the enhanced ESPN Goals app (UK only).