Steven Caulker scored a second-half header against the Swans, who he spent time on loan with in 2011/12, giving the Bluebirds an historic victory on home soil.
After the match, which also saw Swans keeper Michel Vorm sent off late, veteran attacker Bellamy was in a positive mood as Cardiff moved above Swansea and into 12th position on the table.
"It's a big win," Bellamy told Sky Sports. "It's the third south Wales derby I've played in and every single one has been decided by one goal.
"It's always a hard-fought game and today was no exception.
"As well as it being a derby, it's a big three points for us as we really needed it. Next month we've got some really tough fixtures ahead of us.
"It was a home game and we knew it was very important, not just with it being the south Wales derby, but we needed three points as well."
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