Jones was found guilty of "directing insulting and directing abusive language" and "a lack of cooperation and respect" towards the official.
United were also fined £8,900, while defender Jones, 25, was fined £4,450.
His team-mate Daley Blind was also charged with an anti-doping rules violation, but was only fined £4,450.
It is understood Jones was angry because an anti-doping official refused to let him pose for a photo after the game in which the United players held up a banner in tribute to victims of the attack in Manchester on 22 May.
United's 2-0 win over Ajax in Stockholm came two days after 22 people were killed by a suicide bomber following a concert at Manchester Arena.
Jones, who was an unused substitute for the match, will be ineligible for the Uefa Super Cup match against Real Madrid on 8 August in Skopje.
He will also miss the first match of the Champions League group stage.
Blind, 27, played the whole match against Dutch club Ajax in Sweden.
He was charged for "violation and non-compliance" with a rule that states: "Every player designated to undergo as doping control is personally responsible for reporting immediately to the doping control station as notified."
Uefa said that the decisions are open to appeal.
Manchester United's Phil Jones has been suspended for two European games for verbally abusing a Uefa anti-doping officer at the Europa League final.