The Midlands club managed to avoid the drop on the last day of the season after drawing at West Ham United, and according to Smith, survival feels better than when they were promoted last season.
The manager said: "I'm emotional, pleased, proud - all the adjectives to go with it. It's a magnificent achievement.
"My condolences go out to Hayden Mullins and Watford and Eddie Howe at Bournemouth. I can only think about how they feel now. Eddie is a fantastic manager, coach and person.
"But it had to be two out of three go down and I am proud of a magnificent achievement by Aston Villa.
"It feels better than getting up through the play-offs last year. We will now have a few drinks and celebrate."
Smith also thinks the coronavirus lockdown helped Villa to re-organise themselves.
Speaking to the BBC, he shared: "Up to the pandemic, a lot of people had us relegated. We have had to work extremely hard.
"I thought we used the pandemic really well. We've been solid defensively, we have looked strong and managed to stay in the Premier League. It is something to build on.
"We sat down in Minneapolis [on a pre-season tour] over a year ago and came up with three words between us as to how we wanted to be known - and they were: togetherness, prepared and relentless.
"It takes a long time to build a culture but we are growing into that."