Gus Poyet's men looked doomed to relegation after a 1-0 home defeat to Everton on April 12, but a run of seven points from three matches against Manchester City, Chelsea and Cardiff City have revitalised the team.
Sunderland now sit 17th, above the drop zone on goal difference, having played a game fewer than all of the sides in the bottom three.
Cattermole and his team-mates travel to Manchester United on Saturday, before home games with West Brom and Swansea City, and the 26-year-old thinks as little as three points from those three fixtures could ensure survival.
"A few games ago, we were looking to get 10 points," he told the Sunderland Echo.
"After the last three games, we're three points away from that now. Another win might be enough, but we have to look to win every game.
"I know the next game is Man United, but we've done well away from home all season.
"For what's at stake for us, I think we're as committed as anyone at the minute.
"Whether that's against Man United under a new manager or Cardiff, we've got to match everyone in every department."
