The Black Cats drew 2-2 away to Manchester City before beating Chelsea 2-1 at Stamford Bridge to lift them off the foot of the table, with their 4-0 win over Cardiff seeing Gus Poyet's side climb out of the bottom three.
Their draw at the Etihad Stadium ended a run of five straight defeats while the win at Chelsea provided their first Premier League victory in 11 outings.
Goal difference is currently keeping Sunderland out of the drop zone and Cattermole is hoping to draw inspiration from their impressive last two results on the road when they travel to Old Trafford.
"We've been to Manchester City and done well and Chelsea and done well, so why not at United too?" he said to the Sunderland Echo.
"To be honest, going to Old Trafford is not something I think we're worrying about - we're just concentrating on getting out there and doing our best.
"It was strange after the Cardiff City game that there was no massive celebrations in our dressing room afterwards - we were all quite quiet and I think that was because we're so focused at the moment on what we're trying to achieve.
"Yes, we had a great win over Cardiff, but that's just one step and we've got more to do.
"Now we want to go on and try to win the games against Man United, West Brom and Swansea."
Sunderland are in a favourable position to stay up with a game in hand over 18th-placed Norwich, but prior to drawing at the Etihad, the Championship appeared to beckon as they were seven points adrift of safety following defeats to Tottenham and Everton.
But Cattermole believes that the Black Cats have used this to their advantage and are now starting to look up rather than down.
"I think sometimes you have to hit rock-bottom before you can respond," he said.
"That game was the point where we turned to each other and said if this is how it is going to be, it will be a hard run-in.
"We resolved that it could not happen again and we came out fighting.
"We've shown some belief, players have hit form and that was exactly what we needed.
"You need a few heroes to step forward, and I think that's what we've had in recent weeks.
"It has been an unbelievable response when you think where we were at after that game at Spurs, and how we've managed to dig ourselves out.
"We're in a position now where it is in our own hands, and that's all we could hope for."