Anders Dreyer netted the first goal for the Danish rookies in the group phase, but they remain bottom without any points.
Ajax's preparations had been severely disrupted as they were forced to leave behind key players, suspected of being infected with COVID-19, when they travelled to Denmark on Monday.
Yet the likes of first-choice goalkeeper Andre Onana and captain Tadic, were later allowed to join up with their team mates.
Ajax, citing privacy laws, had refused to provide individual medical details and it was not clear until hours before kickoff whether they would be able to field a full-strength side.
Yet the pre-match confusion was quickly forgotten as Antony was set up by Tadic, allowed time to steady the ball and turn onto his left foot to rifle home inside the opening minute.
The Dutch club were 2-0 up in the 13th minute when goalkeeper Mikkel Andersen picked up a back pass, conceding an indirect freekick inside the six-yard box.
Midtjylland lined up every player on their goal-line but were still unable to block Tadic's blast from point-blank range.
After their horror start, Midtjylland were quick to pull a goal back when Dreyer was allowed plenty of room to drive the ball home from just inside the Ajax penalty area in the 18th minute and continued to look dangerous throughout the game.
Yet Ajax went closest to scoring again when Antony had the ball in the home net in the 65th minute but his effort was ruled out for offside after a VAR check.