With the aggregate score locked at 2-2 after extra-time, Forest won the shoot in Denmark to reach the quarterfinals.
Steinlein said afterwards: "I am incredibly disappointed that I will not be going to Istanbul (with the club in the final) on May 20.
"It really upsets me because that's what I had hoped for and dreamed of. On the other hand, I am incredibly proud of the performance that (Mike) Tullberg and the players have delivered. It has been Midtjylland's greatest European adventure and a sign that we have taken the next step as a club.
"I hope that when we look at it from a distance, the boys have huge pride that we are where we are today.
"Most of all, the players have delivered a really great performance. We came third in the (table)... an excellent effort. We have played 16 games, winning 12 of them.
"I actually think we are one of the teams in Europe that has delivered the goods, so to be eliminated by the smallest possible margin is a shame."