The turning point of the match unfolded in the 70th minute when Nemanja Matic was sent off for pushing Ashley Barnes to the ground in response to a nasty one-footed lunge from the forward.
Mourinho insisted that Barnes should not have been on the pitch to commit the studs-up foul as his 31st minute knee on Branislav Ivanovic was worthy of a red card.
The Chelsea boss also disagreed with Atkinson's decision to reject penalty appeals when Michael Kightly handed the ball in the 34th minute and Jason Shackell barged over Diego Costa in the 44th minute.
Ivanovic opened the scoring for the Premier League leaders before Ben Mee headed home an 81st-minute equaliser.
Mourinho, mindful of avoiding another run-in with authorities, chose his words carefully as he criticised Atkinson.
"This game had four crucial moments - minutes 30, 33, 43 and 69
This is the story of this game
I can't comment because it's difficult for me to not say the truth," Mourinho said.
"I'm making it easy for you because if you look at these moments you know exactly what I think about the game.
"What happened to Matic was in minute 69 and minute 69 had a big relationship with minute 30 because normally that player, if I can call him a player, should have been in the shower in minute 31.
"There wouldn't have been a minute 69 if the person in charge had dealt with minute 30 properly."
When asked about Barnes' tackle, Mourinho replied: "I'll leave it with you
I'd prefer not to call it because if I call it I'm going to use some words.....
"A couple of weeks ago when I was here in the press conference after the Liverpool game in the Capital One Cup, the television subtitles had gone on and on about Diego's crimes
Well compare Diego's crimes with what happened against Burnley."
Mourinho may have been guarded with his thoughts, but Chelsea's official website later posted a story which contained no quotes but a breakdown of the incidents.
It read: " MINUTE 30: Branislav Ivanovic is felled by a challenge from Burnley's Ashley Barnes [pictured above], who escapes punishment; MINUTE 33: The Blues are denied a clear penalty as Ivanovic's shot strikes the arm of Michael Kightly inside the penalty area; MINUTE 43: A second clear penalty appeal is turned away, this time after Diego Costa was pushed to the ground by Jason Shackell; MINUTE 69: Barnes goes through Nemanja Matic with a potentially leg-breaking tackle, a challenge which ultimately led to Matic's dismissal."
Mourinho claimed in January that there is a "clear campaign'' by "people, pundits, commentators and coaches from other teams" against Chelsea and the Portuguese hinted that Saturday's events against Burnley have only solidified his belief
"Have I seen a run of decisions like this? I don't remember
I'm happy that I'm not stupid and I understood everything a couple of months ago
I'm not in the mood to laugh," he said.
"If this story that started a couple of months ago finished today with 12 matches to play with an advantage of five points, I'd tell you we will be champions
But I don't know if the story ends here, or if there is more waiting for us."
Burnley boss Sean Dyche refused to antagonise Mourinho, though clearly disagreed with his belief that Barnes should have been sent off.
"I've only seen it as it happened
It looked like a coming together at an odd angle
Then Matic reacts in the way he did," Dyche said.
"I will look back at it again, but I haven't seen it since
But there's more to talk about than that because it was an excellent performance.
"Barnes was involved in something earlier in what sense? Playing football? A charge in the back? Is there anything else? The grass was too short?
"I have total respect for Jose Mourinho, his players and Chelsea
Fantastic manager, fantastic club."
Source : PA
Source: PA