Firmino was booked for shoving Ander Herrera after the forward reacted angrily to the Spaniard pulling his shirt as the visitors looked to counter-attack in the dying embers of a pulsating clash.
Both managers, who became embroiled in a heated touchline exchange, ultimately had to settle for a point as Zlatan Ibrahimovic headed an 84th-minute leveller after James Milner had clinically dispatched a first-half penalty.
The draw ended United’s nine-match winning streak and left them sixth in the table, four points adrift of a Champions League spot, while Liverpool returned to third and lie seven points behind leaders Chelsea.
“He [Mourinho] wanted the minimum of a yellow card, I don’t know,” said Klopp. “I think the ref whistled before anything else happened.
“Roberto is a footballer from head to toe and he wanted to stay in the game. He could have passed the ball but that was a yellow card for Herrera and nothing else.
“In the end it was a yellow card for the guy who wanted to play football. It would have been even worse if someone wants to see it again and all that stuff. We [him and Mourinho] could not have the same opinion in this moment.”
Mourinho denied he was trying to lobby for Firmino’s dismissal. He said: “He [Klopp] thought I was asking for his player to be sent off, I wasn’t. There was no problem at all.”
Klopp also denounced United for an inhospitable welcome, adding: “I read the programme notes and it is the first time that there was nothing about ‘welcome to Liverpool’ or something like this.
“Then I wanted a coffee and in the can was only tea. It’s started already, yeah? It’s all okay. It’s football.”
Mourinho, meanwhile, whose side was chastised for adopting supposedly ultra-defensive tactics during the goalless draw in the reverse fixture at Anfield in October, was happy to point the finger of negativity in Klopp’s direction.
“We changed the qualities of our game and went to a more direct approach because Liverpool were defending with so many bodies behind the ball,” added the Portuguese.
Liverpool opened the scoring on 27 minutes when Milner converted a spot-kick after Paul Pogba inexplicably handled as he challenged Dejan Lovren.
But United levelled with six minutes remaining as Ibrahimovic stooped to divert an Antonio Valencia cross over goalkeeper Simon Mignolet after Marouane Fellaini’s initial flick had ricocheted off the post.