A double from Mexican forward Carlos Vela secured a 2-0 second-leg playoff win and a 4-0 aggregate victory for la Real over Ligue 1 outfit Olympique Lyonnais to see them qualify for the group stages of the illustrious continental competition.
It is the first time the San Sebastian based club have made it in to Europe's premier club tournament since the 2003-04 season, and Arrasate - appointed to the Sociedad hot seat in June - said the feeling among his players at the resounding two-leg victory was invigorating.
"You just have to go into the dressing room to realise how happy we are," Arrasate said after the match on Wednesday.
"It was one of the main objectives in this season and we achieved it.
"We are pleased and we have to enjoy ourselves now, but tomorrow we will start to think about Atletico Madrid because unfortunately time is moving on and we have to carry on."
Arrasate admits he was caught cold by some of Lyon's tactical approach on Wednesday but believes his side could have made the tie even more comfortable in the end.
"They surprised us at the beginning of the game because they played directly, they wanted to play in our part of the field and it was tough for us," the 35-year-old said.
"But we managed to withstand the pressure and as time went on we created lots of chances to score. At the end of the day, I think the score is even less than we deserved.
"It is a magic night for us. We were aware we were playing for an important qualification and we couldn't fail.
"The happiness is for everyone, part of this success is for the supporters as well. We are very pleased to have won and to be able to celebrate with them because they deserved it too."
Real Sociedad are seeded in Pot 4 for Thursday's group stage draw in Monte Carlo, which begins at 17:45CET.