He said: "We don't have to be worried about this run and let any fear come into our play.
"It is great to be at Sunderland. It's a dream for me to be back among the big boys and making my debut at Old Trafford said everything about why this is the best league to be in.
"We can be ambitious for the rest of the season and I'm really looking forward to it."
On the defeat at Manchester United, Higginbotham also told the Sunderland Echo: "The way we lost was the biggest disappointment because we restricted a very dangerous team to efforts from outside the box for most of the game.
"One lapse was the difference. All the lads were gutted to concede like that, but we gave everything we had and we can keep our heads up.
"Craig Gordon made some good saves at times, but I thought every man out there performed well and on another day, we might have got something.
"That's the Premier League, that's the quality you are up against and you are not going to get away with it at this level."