At least two windows were smashed after objects were thrown at the bus as it was surrounded by a large group of fans outside the stadium, before police were able to intervene.
There were no reported injuries among the Atletico Madrid party, and the team were able to begin their round-of-16 match on time. Sociedad won the match 2-0.
Video footage showed Atletico coach Diego Simeone standing at the front of the bus, appearing to remonstrate with fans and police.
Asked after the match about the incident, Simeone said: "You all saw what happened outside. We were en route, alongside the Real Sociedad fans. I imagine they're always in that position. Suddenly the [police] protection wasn't there. You saw that.
"But it didn't affect the game at all. When the game started I said to [Real Sociedad coach ImanolAlguacil] that it had absolutely nothing to do with him, but he should send my regards to whoever had to prepare, and didn't prepare, whatever that was."
The incident follows the suspension of the derby between Real Betis and Sevilla on Saturday, also in the Copa del Rey, after the Sevilla player Joan Jordan was struck by a plastic pole thrown from the stands. The Spanish football federation announced on Wednesday that Betis' Benito Villamarin stadium would be closed for two matches as a punishment for that incident.
Real Sociedad had tweeted on the morning of Wednesday's game with Atletico, calling for "respect and tolerance."
On Thursday, La Real released a statement on their Twitter page which read: "Real Sociedad regret and condemn the incidents which occurred yesterday upon the arrival of the Atletico Madrid bus and around the stadium.
"The uncivilised behaviour of a minority outside the stadium does not represent the true fan base which, throughout the entirety of yesterday's encounter, showed exemplary and sporting behaviour.
"Real Sociedad once again reject any act of violence and will continue to promote its sporting values."
There has been animosity between segments of the two teams' fanbases since the death of a Real Sociedad fan, Aitor Zabaleta, who was stabbed near Atletico's former Vicente Calderon stadium in December 1998.
Spanish media outlets reported on Wednesday that in addition to the bus attack, an Atletico Madrid fan had suffered a head injury after being hit by a glass.