Sugar tired of Apostolova's sob stories in the boardroom and axed the Bulgarian following a disastrous printing task for the girls' team.
The opening challenge of the series involved the two teams buying £500 of blank materials, creating designs and printing them, and then selling them to the public for the largest profit.
Apostolova was part of the girls' sub-team, who were sent to London Zoo to flog their merchandise. However, the sub-team's bickering, aggressive sales approach and Apostolova's poor North London geography skills appeared to cost them time and money in the task.
When the important numbers were added up in the boardroom, Nick Holzherr's Team Phoenix triumph with a £616.20 profit. The girls' Team Sterling, who were project managed by Gabrielle Omar, only collected £214.80. The boys were rewarded for their efforts with canapés and cocktails at the house.
Back in the boardroom, Lord Sugar was quick to pick up on the failings of the sub-team and Nick Hewer cornered the girls on their behaviour with a local shopkeeper.
"The shopkeeper was so affronted by their sales approach that she wouldn't buy anything; she couldn't stand being cornered by a pack of baying hyenas," said Hewer.
Apostolova and Omar were joined in the final boardroom showdown by Katie Wright, who Hewer accused of "going missing" on the task.
Wright defended her minimal work on the challenge, insisting: "I absolutely swear I was not hiding, there's no way I would do that."
Watch the Apprentice girls' team blunders:
Meanwhile, Omar fought back at criticisms of her management skills from Apostolova, ranting: "I have a small voice but everybody respected me, everybody that was under my eye at least listened to me but with you all I hear is excuses, excuses, excuses, what's your excuse for everything that you did?"
Sugar appeared to be leaning towards firing Wright for her minimal impact on the task, but Apostolova's continual chatter and stories about her tough upbringing appeared to alter Sugar's opinion.
"I don't want to bring out the violin and listen to your sob story," he moaned.
Commenting on her gripes about the girls' team's methods, he added: "Bilyana you're sitting there talking like a city strategist, this is a two bob outing. This is not a takeover of Goldman Sachs, this is simple stuff!"
In his final summary, Sugar said: "I'm not listening to anybody anymore now, I've had enough of listening to you. I'm telling you what the situation is whether you like it or not.
"It is a difficult decision to make in this first task so the person leaving today may feel that they haven't had enough time to show themselves but that's tough, that's what this process is about. I did say in my introduction that I didn't want anybody to hide and on that basis..."
After another interjection from Apostolova, he added: "You don't do yourself any favours, Bilyana - you're fired."
In the black cab home, the 25-year-old said: "I feel the decision wasn't fair and I don't think that my performance cost us the task. But my business plan is excellent so I'll just move on straight away."
The Apprentice continues on Wednesday at 9pm on BBC One.