Strachan was frustrated by the way Horwood chose to go with his assistant's flag, which was raised when Tarmo Kink appeared to shove full-back Alan Dunne to the floor just before the half hour, rather than make the decision himself.
The Boro boss walked off the New Den pitch with the officials at full-time, questioning why the penalty was awarded in that manner.
It was not that he disagreed with the controversial spotkick, which was scored by Steve Morison, just that he felt the man in the middle should have awarded it.
It's not a complaint. I get paid more money than my backroom staff to make tough decisions, I don't send my backroom staff in to tell a young player he is leaving on a free, said Strachan.
I do it. The referee gets paid more than the linesman so I think there was lack of responsibility from the referee to go with the linesman.
I tell my players that they must take responsibility.
That's what the referee should have done.
The result hurts but if you give it your best shot then that hurts. It's when you have let people down when it is hard to live with.
Middlesbrough were then left two behind when Liam Trotter added a second six minutes before half-time, which left Middlesbrough effectively out of the Carling Cup despite Scott McDonald's late consolation.
It was a smashing game of football and I am pleased with the part we played in after Sunday's match with Sheffield United, he said.
I will sleep better tonight after Sunday, which was not a good game and we didn't play well. We still want to enjoy our football and I would like to see improvement, but all in all we moved the ball well.
Having been encouraged, Strachan, who has a doubt over the fitness of Nicky Bailey's groin problem for Saturday, wants to see further progress made against Barnsley this weekend.
He said: We have four good players missing at the moment.
We are chopping and changing all of the time. Nicky Bailey played with an injury.
The lads take responsibility and I like that. I got them playing better. You never will know how long it will take. Blackpool thought they had cracked it.
Stephen McManus, Barry Robson and Kris Boyd have all been included in the Scotland squad for next week's European Championship qualifiers with Lithuania and Liechenstein.
Millwall manager Kenny Jackett hailed his side's character after victory over Middlesbrough.
The Lions suffered a 3-1 defeat at Elland Road on Saturday, their first loss of the campaign.
It was a challenge for us today given the Leeds result and a big challenge at that,'' he said. It's an opportunity for us and one we took.
You never want to go on a run of losses, you never want to lose at all and it is important, especially at the start of the season.
And against one of the bigger Championship clubs tonight we did want to equip ourselves well and put on a good performance.'' Jackett backed goalkeeper, David Forde, who also made an error against Leeds, to bounce back.
He made an error, but over the last two years he has been an excellent goalkeeper for us,'' he added.
He has got to put that one behind him now it didn't cost us because he has been a good goalkeeper for us and will continue to be.