United scored a last-gasp leveller to earn a point against the leaders, giving Louis van Gaal arguably his most impressive result since taking charge.
However, even with the late heroics of Robin van Persie, the Dutchman's side are still 10 points off the summit.
But full-back Shaw believes the best is yet to come from the Old Trafford outfit, and the England international reckons the club's rivals should be running scared.
"Of course we are still in the title race," the 19-year-old said. "It's a long season and we are at the start of it. You don't know what will happen during the middle of the season.
"We just have to keep winning the games coming up. There will be tough games but we know we are good enough to win them.
"We know we have still got the best to come. I don't think we are playing our best at the moment and sometimes you saw that against Chelsea.
"It will take time but when we are all on form I think we are going to be a scary team to face."
United's next chance to prove their title credentials will come against champions and fierce rivals Manchester City on Sunday.
Van Gaal's charges make the short trip to the Etihad Stadium knowing a win would take them to within one point of City.
And Shaw is relishing the prospect of his first Manchester derby.
"It's pretty obvious what this match means to the fans and the club," he added. "I don't think I need to be told about it and the other [new signings] will be aware too.
"I've watched it for the past years and I've seen how much it means to everyone. It's one of the biggest occasions in football after the World Cups and Champions League finals.
"It's a massive occasion for everyone involved in it. We need to come out firing on Sunday."
