FC Midtjylland thereby missed the opportunity to reclaim first place in the Danish Superliga, which was overtaken by AGF after Friday's victory over Silkeborg.
FCM has 32 points in second place, while Sonderjyske, thanks to their impressive run of results, are now in fifth place with 25 points.
Sonderjyske did not seem burdened by the fact that FC Midtjylland went into the match unbeaten in over two months in all competitions.
Chances were few and far in the beginning of the match, but the visitors still managed to claim the lead when Aral Simsir drove the ball across the pitch and finished off with a flat, stinging drive into the far corner.
However, FC Midtjylland were hardly done celebrating when the home side struck back when Kristall Mani Ingason carried the ball well into the box after excellent preparation by Alexander Lyng and hammered the equalizer into the net.
A somewhat uneventful half came to life in the final minutes of the second half, when Billing received his second yellow card for swinging an arm at Olti Hyseni, much to Tullberg's frustration.
The home side then capitalized on their numerical advantage in injury time when Lirim Qamili converted a penalty after a lengthy VAR check.
In Farum, Fredericia suffered their eighth successive defeat when they were torn apart by FC Nordsjaelland and lost 5-0. Levy Nene and Justin Janssen were the big heroes of the day for the home side as they both scored twice before Norwegian midfielder Runar Norheim rounded off a superb afternoon for Jens Fonsskov Olsen's side with goal number five, nine minutes from the end.
After their impressive win in the last round against FC Copenhagen, Vejle were sent crashing back to earth when they suffered a 5-2 defeat in Viborg.
Anders Jacobsen otherwise gave the visitors a 1-0 lead after ten minutes, but before the interval, the home side had effectively turned the match on its head as they led 4-1 at the break. Christian Gammelgaard gave the visitors a lifeline in the second half before Mads Sondergaard made it 5-2 shortly before the end.