Curtis urges Swans to "be bold" against United
The Swans are enjoying a three-game unbeaten run in the Premier League which has coincided with three clean sheets in a row - the first time they achieved such a feat since March 2012.
Draws against West Ham and Crystal Palace were coupled with a crucial home win over West Brom on Boxing Day and the Swans now face a Manchester United side that is without a win in eight matches.
Swansea have won their last three games against United and could haul themselves further clear of danger with another victory at Old Trafford on Saturday.
"We've done exceptionally well against them in the last three games," said Curtis. "But we know it will be a tough game and we'll have to be at our best again.
"There's a confidence creeping back into us with the results we've had and the performances have been excellent too.
"Some of their (Manchester United) results have been a little indifferent but they were excellent against Chelsea on Monday and the game could have swung their way on another day.
"We've just got to go there and perform, be bold and play our football."
The Swans moved out of the bottom three with results over the festive period, and head into Saturday's clash two points clear of the relegation zone.
And Curtis reiterated his commitment to continue in caretaker charge until the club finds a new permanent first-team manager.
"You have to prepare anyway because they are all important games," he said. "If we had lost the games there would have been huge pressure on the club to get someone in.
"The performances have been good. We've shown a lot of character too but we've got to ignore what's happened before and focus on now.
"I've always thought we are capable of getting out of it - but it looks like it might be tight. If we carry on like we have been, confidence should grow and hopefully we can start moving up the table and pull clear."
Curtis also confirmed that Wayne Routledge should return to the squad after missing the draw with Crystal Palace, while Federico Fernandez is hopeful of being fit after recovering from a bruised heel.