Besic was sent off in the 20th minute at the Riverside Stadium, with referee Simon Hooper showing a straight red card after the Bosnian midfielder pulled Bradley Dack's shirt to prevent his opponent breaking clear into the box.
Hooper initially looked to have missed the incident but after a pause when he appeared to receive information via his earpiece, he gave Besic his marching orders.
Pulis accepts Besic tugged Dack's shirt but reckons goalkeeper Darren Randolph would have got to the ball ahead of the Blackburn player.
He said: "The referee doesn't send him off - it's the fourth official and the linesman on our side. I'll say it now - Besic does pull his shirt.
"But I believe our goalkeeper was getting to the ball first before their lad. I think it's a yellow card, not a sending off.
"How they make that decision from that distance - and a decision that ultimately changed the game - is beyond me. I think you've got to be 110 per cent sure that you're stopping him from scoring a goal, or stopping a clear-cut chance."
Boro suffered a double punishment, with Charlie Mulgrew curling home Blackburn's opener from the free-kick that followed Besic's departure.
At that stage, the Teessiders looked like suffering a second successive home defeat, but Pulis' bold half-time changes altered the course of the game, with substitute Britt Assombalonga curling home a magnificent equaliser midway through the second half.
Pulis added: "The second half was good. I just thought the sending off completely knocked the stuffing out of the lads.
"It was disappointing for Fletch [Ashley Fletcher, who was substituted after Besic's dismissal] because we wanted to play him a little bit on the left-hand side, cutting in, and I think that was working quite well up until the sending off.
"Bringing Lewis [Wing] on, I decided to play him and Jonny Howson to try to get around Jordan [Hugill] as much as they possibly could.
"When we got the ball, at least we were playing with three players up there, not Fletch just coming off that one side. Then in the second half, you bring Britt on and you take a chance."
Blackburn manager Tony Mowbray was disappointed at his side's failure to hold on to their lead in the second half.
The former Boro boss felt his side had created enough opportunities to put the game to bed, but praised his former club's spirit as they battled back with 10 men.
Mowbray said: "It's a very frustrated dressing room. We shouldn't have dropped points, playing against 10 men and having scored from a free-kick.
"Ultimately we needed the second goal, and it didn't come. I think if there's one team in this league you don't want to play against with 10 men, it's probably Middlesbrough with how good they are behind the ball.
"They're so defensively stable and solid, and they did what they had to do. I thought we created some chances, Bradley hit the post and we had a few shots.
"We got into some decent areas, but probably lacked that last bit of quality to pick the right pass and put the ball in the back of the net.
"Credit to Middlesbrough. I was right behind the finish, it started a yard or so outside the post and he brought it back in.
"It was a great finish, but we're hugely frustrated. Middlesbrough did what they do."