Pierre-Emerick Aubameyang's hat-trick and a second-half goal from Alexandre Lacazette inspired the Gunners to a memorable triumph at Mestalla and convincing 7-3 aggregate success.
Reaching the top tournament via the Europa League has become the club's likely last resort after a run of one win and four defeats in six top-flight matches left them three points adrift of the Premier League's top four with one game to play.
"We had confidence," Emery told reporters in Thursday's post-match news conference.
"We had two possibilities – the Premier League we lost, now we are in the Europa League.
"After the Premier League was lost, we took a strong stance in our mind to prepare, to focus, to work hard to take this feeling.
"We can be proud of the players, of the work, of our supporters.
"It's not finished. We have one step more to play in the final. We are happy but we know it will be difficult to win."
Aubameyang and Lacazette have now combined for an impressive 48 goals in all competitions this season.
Emery commended the pair on complementing their prolific goal-getting with defensive diligence.
"I am proud of them," the three-time Europa League winner said. "Big players can be good strikers but when their commitment is like today, defensively also, to work to help us, it is amazing.
"Before the match, we spoke [as a team] at the hotel. We talked about our moment, our experience, our being together, to make the best performance. They did that."