The violence broke out after a drone dangling an Albanian banner circled over the field during the first half of Tuesday night's game. A Serbian player grabbed the banner as Albanian players tried to protect it, and the game was called off before halftime amid the chaos.
The case was quickly assessed by UEFA's disciplinary unit on Wednesday after receiving reports from the delegate, match officials and security.
The Serbian hosts have been charged over "insufficient organization," fireworks being set off, missiles being launched, a crowd disturbance, a field invasion by supporters and the use of a laser pointer, UEFA said.
The Football Association of Albania has been charged for "refusing to play" and over the display of "an illicit banner," the UEFA statement said.
The case will be dealt with by UEFA's control, ethics and disciplinary Body on Oct. 23.
Serbia could have to play its next Euro 2016 qualifier at home on Nov. 14 against Denmark without any fans.
UEFA President Michel Platini said he was saddened by the violence.
"Football is supposed to bring people together and our game should not be mixed with politics of any kind," Platini said. "The scenes in Belgrade last night were inexcusable."
There was also condemnation from FIFA President Sepp Blatter, who wrote on Twitter: "Football should never be used for political messages. I strongly condemn what happened in Belgrade."
Albanian fans had been warned against attending the game between the two Balkan rival nations that have had turbulent relations — mainly over the former ethnic Albanian-dominated Serbian province of Kosovo that declared independence in 2008.
But neither Serbia nor Albania asked to be separated in qualifying, said Pinto, who also highlighted the lack of ongoing military conflict between the nations.
Armenia and Azerbaijan, which have a territorial dispute, have been kept apart by UEFA in draws since their scheduled Euro 2008 qualifiers were cancelled, and Spain refuses to play new UEFA member Gibraltar because of a sovereignty dispute with the British territory.