Mohamed Salah became Liverpool's leading scorer in the Champions League when he pounced on a loose back-pass to run through and poke the ball through the legs of the goalkeeper after just 55 seconds.
It was his 22nd goal in the Champions League, one more than Steven Gerrard.
Already qualified as Group D winner, Liverpool rested a number of key players – though Salah played the whole game – and Midtjylland took advantage by taking the game to the English champions in the second half.
After Evander struck the crossbar with a fierce shot, Alexander Scholz earned Midtjylland their second point of the first group-stage campaign by converting a penalty following a foul on Anders Dreyer by Liverpool goalkeeper Caoimhin Kelleher.
Both teams had goals disallowed in final minutes, Midtjylland for offside and Liverpool for a handball against Sadio Mane.
Midtjylland finished in last place in the group, missing out on a Europa League spot.
The awarding of the penalty was contentious, with referee Francois Letexier having blown for a free kick for offside as Dreyer was brought down. VAR advised the official to go to the pitchside monitor and the Frenchman subsequently awarded a spot kick from which Scholz scored.
Jurgen Klopp selected Liverpool's youngest Champions League starting lineup, with an average age of just 24 years and 26 days, though Salah (28) and Fabinho (27) were in it.
While Fabinho was replaced at halftime by Billy Koumetio, who became Liverpool's youngest player in the European Cup or Champions League at the age of 18 and 25 days, Salah stayed on until the end.
It was a surprising decision by Klopp considering Liverpool has three games in a seven-day span starting on Sunday (Monday NZT).
Trent Alexander-Arnold was captain for the night and became the club's third-youngest skipper, and its youngest in a European game.
Another academy graduate, Leighton Clarkson, made his European debut having previously only featured in two domestic cup games last season.