The former judge, who went on to put Waissel through to live shows, made the confession in her new autobiography Cheryl: My Story.
Waissel, who courted controversy throughout her stint on The X Factor, famously forgot the words to Queen's 'We Are the Champions' when she first tried out for the show in 2010, before singing Etta James's 'At Last'.
"Simon [Cowell] had spent the past two years drumming into me that we needed acts who would be 'good TV'," Cole confessed.
The 'Call My Name' star took a break from the series while she battled malaria, and later saw Waissel perform at Judges' Houses as part of the Girls category.
"When I saw Katie perform this time, I had to admit that she was... quirky, intriguing and busting to succeed.
"In short, she had the character and drive it took to withstand the pressure of the show, and so I put her through, even though she messed up when she sang in front of Will [will.i.am]."
Speaking of Cher Lloyd, who also competed that year, she added: "I absolutely loved her and had done from the start. She reminded me of myself, coming from a similar background and being gutsy, putting herself out there.
"She basically broke down when it was her turn to sing, but I still put her through because I believed in her and I felt sure she'd blossom as the competition went on."
Lloyd has previously revealed that Cole didn't keep in touch after The X Factor, quipping that she had "more chance of winning the lottery" than catching up with her.
The pair fell out publicly after Lloyd joked that she would only perform with her former mentor if she sang live without the aid of Auto-Tune. Cole warned her to "be careful who you kick on your way up" and said she "needs to learn a lot".
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