Wenger's first XI for the Capital One Cup third round contest at the Hawthorns featured nine adjustments from Sunday's Barclays Premier League win over Stoke and several players inexperienced at senior level.
One of those, 20-year-old first-time starter Thomas Eisfeld, opened the scoring in the 61st minute with a slotted finish before Saido Berahino headed a 71st-minute equaliser.
The contest finished 1-1 after 120 minutes, with Albion substitute Morgan Amalfitano having hit the bar in a strong conclusion to extra time for the hosts.
The Baggies then initially had the advantage on penalties after Serge Gnabry's effort was saved, but fellow Arsenal teenagers Kris Olsson and Chuba Akpom subsequently netted theirs and Craig Dawson and Amalfitano missed for West Brom, allowing Nacho Monreal to seal a 4-3 shoot-out triumph for the visitors.
Wenger - whose team host Chelsea in round four - said: "I am proud. We played with many young players and they showed they are not only good footballers but that they can fight as well.
"I decided to go for young players with the penalties and they coped well with the pressure.
"Of course, penalties are a bit of a lottery, but overall I think the most important thing is that we played well in the game and that the young players came out and showed that they can play the way we want to.
"It is always good to know that you can win even when you thought you were dead and tired.
"If you dig deep and in the end you are rewarded, that is a very important lesson for a young player.
"We kept going forward and trying to win the game until the end and that is very positive."