The Belgian goalkeeper - Liverpool's last man - brought down Mame Biram Diouf on the edge of the box with the striker winding up to shoot with the visitors leading through Sadio Mané's effort.
But Atkinson showed Mignolet only yellow and Liverpool went on to win 3-0 as second-half substitute Mohamed Salah took his tally to the campaign for 17 goals in all competitions with a late double.
While Stoke boss Mark Hughes felt his side were hard done by by the referee's first-half decision-making, Klopp insisted, contrastingly, that Mignolet's foul did not warrant a sending off.
Liverpool win was "well deserved"
"It was hard work, but that’s how football should be, especially when you go to a place like Stoke. It was no surprise," he told his post-match press conference.
"I think we could have made it a little bit easier for ourselves in the first half if we'd used the two fantastic chances we had. They were not lucky or anything; they were really brilliant, brilliant play.
"We didn't use them, so we needed a little bit of luck in the situation around the yellow card for Mignolet.
"I don't think it was a red card, but of course they could have scored in that situation and then the free-kick in this situation from [Xherdan] Shaqiri was another real threat.
"Second half, still an open game. We should have been more clearer in different situations with better passing in this moment, or a little bit cooler in another situation.
"But of course the atmosphere was there and Crouchy as well, so that means you cannot defend these balls. Only after winning the ball, we should have done a little bit better.
"Then we changed twice and could decide the game. Two fantastic goals, brilliant play – especially the second goal and the third one was a nice counter-attack and a nice finish. In the end, it was well deserved but it was hard work. No problem with that."
Klopp delighted with Salah and Mané's contributions
Salah stretched his outstanding return to 12 goals in his first 14 Premier League games since joining the club from AS Roma in the summer despite playing only 27 minutes.
Salah has now netted in all of his last five league games and scored braces in three of his last four in the Premier League, with two more than any other player in the top-flight at this stage.
On the Egyptian's impact and continued goalscoring form, Klopp continued: "Obviously that's good for us.
"It is difficult for all of the players coming on with this temperature.
"Last week [against Chelsea] we had a little bit of a problem [with the players that came on] and the next day more problems than the players that started.
"Obviously sitting on the bench is not too nice, but today it worked really well. Two fantastic goals."
Likewise Mané - who only came off the bench to play for six minutes in the draw with Chelsea at the weekend after being benched - was excellent and in addition to his goal provided an assist for the first of Salah's two goals.
"Good, good, good. Very important for us," hailed Klopp on the Senegalese winger's performance.
"He needs rhythm now. He was in perfect rhythm, then break. Good rhythm, then break. And now he needs a little bit of time. That’s no problem.
"He was really happy and it was absolutely fine with him, we all know in one or two situations if he is in a click-moment then he can score as well. It's all good, he is ready to work. He is very important for us."