The fashion brand was criticised by one of its own models, Liz Kennedy, in a long post on Instagram in which she said: "Suicide is not fashion."
"Let's not forget about the horrifying history of lynching either," she added.
Burberry boss Marco Gobbetti said the brand was "deeply sorry for the distress" caused, adding: "It was insensitive and we made a mistake."
The hoodie featured in a collection called Tempest which was on the runway at London Fashion Week on Sunday.
The show featured "rebellious youths" scaling walls in one space and Burberry's creative director Riccardo Tisci dedicated it to "the youth of today".
Model Liz, who had been part of the Burberry show but didn't wear the hoodie, said the design was "not glamorous nor edgy".
She wrote on Instagram: "How could anyone overlook this and think it would be okay to do this especially in a line dedicated to young girls and youth?
"The impressionable youth. Not to mention the rising suicide rates worldwide."
Liz said she felt "extremely triggered" after seeing the design and felt "as though I was right back where I was when I was going through an experience with suicide in my family".
She told her Instagram followers she had tried to bring up the issue in the dressing room but was told to write a letter instead.
Liz added: "I had a brief conversation with someone but all that it entailed was 'It's fashion. Nobody cares about what's going on in your personal life so just keep it to yourself'.
"Well I'm sorry but this is an issue bigger than myself. The issue is not about me being upset, there is a bigger picture here of what fashion turns a blind eye to, or does to gain publicity.
"A look so ignorantly put together and a situation so poorly handled."
Burberry boss Marco Gobbetti said Liz's experience "does not reflect who we are and our values".
"We will reflect on this, learn from it and put in place all necessary actions to ensure it does not happen again."
Riccardo Tisci also apologised and said that "while the design was inspired by a nautical theme, I realise that it was insensitive".
Burberry is the latest fashion brand to be criticised for releasing designs which have been considered insensitive this month.
Gucci pulled a woollen jumper from sale after it was criticised for "resembling blackface".
Katy Perry also removed two items from her shoe range after complaints that one of the designs was racist.
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