The Reds ace netted within the first minute of their final Group D clash to claim his 22nd strike in the competition while with the club.
His effort saw him overtake legendary midfielder Steven Gerrard's tally, while it was also the fastest goal they have ever scored in the competition.
Salah was a surprising starter given manager Jurgen Klopp's frustration with the current hectic fixture list, and the fact they have already qualified for the knockout stages
The Egyptian was one of only a few first-team regulars playing in the dead rubber, including Fabinho, Naby Keita, and Divock Origi, but he made the most of his chance.
It was also a big match for young goalkeeper Caoimhin Kelleher and midfielder Leighton Clarkson, who is well thought of by the Reds coaching staff, while Trent Alexander-Arnold was named captain for the night.
But it was Salah who got them off to the perfect start in Denmark.
Any hopes their opponents had of getting a shock win were dashed in just 55 seconds as a calamitous pass from Alexander Scholz had Salah bearing down on goal, after beating Erik Sviatchenko to the ball.
Dion Cools couldn't stop the Egyptian either and he rolled home past Jesper Hansen to create history twice with one goal.
Liverpool didn't have it all their own way in the rest of the first half with Midtjylland looking handy down the right, winning several corners, but failing to take any of these chances.
One set-piece found Sviatchenko in acres of space but the ex-Celtic man was never in the right position to make the most of it and gently nodded the ball wide.
Origi showed just how rusty he is, bobbling an effort across the face of goal when in reality he could have taken the ball on in the penalty area before shooting.
And Fabinho had to be alert when he cleared Sory Kaba's header off the line following more good work on the right.
Youngster Billy Koumetio was brought on for the Brazilian at half-time, becoming their youngest ever Champions League participant, aged 18 years and 25 days, and had a busy opening few minutes.
He dealt with a couple of threatening attacks, while another shot from the Danes hit the crossbar as they looked to rock Liverpool.
There was also more injury woe for manager Klopp with left-back Kostas Tsimikas, a backup to Andrew Robertson, picking up a calf issue before being taken off.
To make matters worse, the Reds then conceded a penalty when VAR helped referee Francois Letexier confirm Kelleher brought down Anders Dreyer in the box.
And Scholz, atoning for his earlier error, fired into the bottom left corner.
Klopp sent on the cavalry with Jordan Henderson and Roberto Firmino also taking to the field as they attempted to turn tide, having watched the Danes dominate.
Midtjylland thought they had got themselves a second after 75 minutes when, following a scramble, Scholz managed to fire in from a tight angle but, after a lengthy VAR check, it was ruled out for offside.
Try as they might, the Reds struggled to get a foothold in the game and were restricted to tame counters via long balls forward.
Another Sviatchenko header brought a great save from Kelleher but the resulting Liverpool attack was halted by a poor Jota cross.
And just when it looked like Liverpool had claimed a last minute winner through Minamino, the goal was ruled out thanks to another frustrating VAR intervention.
This time a lengthy check saw handball cancel it out, despite it initially being chalked off, incorrectly, for offside.
Eight minutes of injury time were tacked on to the eventful 90 minutes but neither side were able to get the winner they were hunting for.
Liverpool forward Mohamed Salah has become the club's leading goalscorer in Champions League history as they drew 1-1 with FC Midtjylland.