Man Utd started the game brilliantly and wasted numerous chances to go in at half time more than a solitary goal to the good.
Cristiano Ronaldo was one of the culprits as he missed a penalty shortly before Jadon Sancho's opener.
After the break they saw Matt Crooks equalise for the Championship side, although they continued to miss chances themselves with Bruno Fernandes particularly wasteful.
It finished 1-1 and extra time could not separate them, and all that remained was for Anthony Elanga to miss the decisive penalty to give Boro an 8-7 shootout win.
"We gave away too many chances, especially in the first half," United interim manager Rangnick said.
"We should and could have easily been up 3-0 after 45 minutes and even in the second half we missed some great opportunities, even after they scored their goal that shouldn't have stood.
"But Bruno's chance was almost like a penalty as well, so we have to blame ourselves for giving away and wasting too many opportunities in the game.
"And, in the end, if it's on penalties it's also a question of luck and in the end we lost that game and we are all incredibly disappointed."
United seemed to visibly tire as the match went on, which left some fans questioning the wisdom of allowing Edinson Cavani and Jesse Lingard extended leave.
Rangnick, though, has defended those decisions, explaining: "No, because [Cavani] played for Uruguay on Wednesday.
"With the jet lag and the travel arrangements, I don't think it would have made sense. He would not have been in the squad anyway.
"And with regard to Jesse, I told you what my information was and my knowledge was, and of course I would have loved to have Jesse in the squad.
"We were one outfield player less on the team sheet today, so why should I have allowed him to have four or five days off?
"I would have loved to have had him in the squad and be in the squad, but it was not the case and therefore it doesn't make sense to speak about players who were not available."