The visitors grabbed the initiative straight from the start and were rewarded after 23 minutes of play when a delivery from Sofus Johannesen took a deflection via Junnosuke Suzuki to leave FC Copenhagen keeper Runar Alex Runarsson helpless.
Bad then turned to worse for the reigning champions after 38 minutes when Mathias "Zanka" Jorgensen took too long on the ball, allowing Felix Winther to poke the ball for Friday Etim, who slotted the ball home to double the lead for the visitors.
FC Copenhagen coach Jacob Neestrup chose to react from the start of the second half when Mads Emil Madsen and Robert Silva replaced Oliver Hojer and Kenay Myrie, but it did little to help.
FC Copenhagen thought they had cut the deficit ten minutes before the end when Mohamed Elyounoussi capitalized on Fredericia keeper Etienne Green failing to hang on to a shot from distance, but Elyounoussi was adjudged to be in an offside position.
In injury time, the hosts managed to cut the deficit when Youssoufa Moukoko converted a penalty after Svenn Crone had brought down Viktor Dadason in the box and subsequently was sent off.
But it was too little too late for the hosts, who are now left in 8th place, six points from the relegation zone in the Danish Superliga.
In the other match of the relegation playoffs on Sunday, Silkeborg IF recorded their first win this year when they defeated Randers 3-0 on away ground. .
Against expectations, it was the visitors who dictated play in the first half, and they were duly rewarded when they scored twice within five minutes from the 28th to 33rd minute to take a firm grip on the game.
First, Tonni Adamsen directed the ball into the net from a fine delivery from Sofus Berger, and then shortly after, Adamsen turned provider for Callum McCowat, who scored from the edge of the box with a stinging drive.
Randers coach Rasmus Bertelsen made three substitutions from the start of the second half, and even though the home team improved slightly, they never got the goal that could have turned the tide in their favour. Instead, Silkeborg put the final nail in the Randers coffin 16 minutes from the end when Andreas Poulsen scored with a terrific long-distance shot.