The Bayern Munich man and his international team-mates had been talked up prior to the tournament, but they finished bottom of Group F, losing two of their three games – including against supposed minnows Iceland.
Now, however, Alaba has described his disappointment and defended his side.
"I know you're disappointed – and believe us, so are we," he wrote to his one million Twitter followers.
"We worked so hard for this tournament, each and everyone of us."
And the 23-year-old stressed Austria's commitment throughout the competition, regardless of results.
"We gave it all we had for you, you gave it all you had for us," he added.
"To each and every one of us it's a huge honour to play for his country.
"For now we have to deal with the situation and slowly begin to prepare for the next tournament."