Boro are second from bottom with four matches left to play, and are six points behind 17th-placed Hull.
Three of Boro's four remaining games are against current top-four teams.
"We've got that one over the line," Gibson told BBC Tees. "Why can't we beat Manchester City this weekend?"
"We got that win, we're confident, even if people think we're down or won't win, whatever. We'll keep doing what we do, believing and fighting," Gibson added.
"That result will give us that lift to go and do that even more. We beat Sunderland, it's only one win, but for us it's absolutely huge."
"It's been an awful experience, it's been the most testing time of my career," Gibson added.
"The season hadn't gone the way we wanted but this was a big win - the derby is a massive game in isolation, absolutely huge."
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