United stretched their unbeaten run to 10 matches on New Year's Day, but they only took a point away from Stoke after Radamel Falcao cancelled out Ryan Shawcross' opener.
Despite the disappointing result, United's standing in the table actually improved.
Chelsea's shock defeat to Tottenham means Louis van Gaal's side are now nine points behind the Blues and joint leaders Manchester City.
Shaw, therefore, feels United's title chances should not be written off.
"We are not going to say the Premier League is over yet," the United left-back said.
"The league is still our number one priority
We want to win that.
"Coming to Stoke is always going to be a tough battle
There's a lot of balls in the air so it's tough.
"We know when we come away we have to get three points so we are not very pleased with the point."
After an injury-plagued start to life at United, Shaw is hoping to make a fresh start again in 2015.
Shaw was dropped from the United team in pre-season as he was out of shape and shortly after his comeback he was out for over a month because of a hamstring injury.
Then, two months after his return, Shaw suffered an ankle injury against Arsenal and was missing for six weeks.
"The main thing for me now is to get a run of games together," the 19-year-old said.
"It was a great feeling watching the team do well when I was out
Now I am back fit I want to be a part of the winning run.
"I had 60 minutes against Stoke
That was pre-planned
If I had been able to go on longer then I would have done."
Ashley Young excelled in Shaw's position during his absence, but the former Aston Villa man suffered a hamstring injury in the 1-1 draw at the Britannia Stadium and is going to be out for "many weeks", according to manager Louis van Gaal.
Shaw thinks he will have to fight hard to retain his place when Young returns.
"There's massive competition for me," the Â£28million defender said.
"I think it's a big credit to Ashley Young
He has been superb, one of our best players for the last four or five matches
It's just unfortunate for him that's he got this injury now
I hope it's not too bad
I hope he comes fighting back again."
Next up for United is a trip to League One bottom side Yeovil in the third round of the FA Cup on Sunday.
Yeovil became known as FA Cup giant-killers during their non-league days when they knocked no fewer than 20 league sides out of the competition.
For that reason, Shaw knows United's stellar squad must not under-estimate their opponents.
"We've seen in the past that the big teams have been knocked out by lower league teams so we know it's going to be a tough game," Shaw said.
"We're going to focus on that now because we want to go on a good cup run
That's the reason I came to Manchester United - because I want to be winning trophies."
Source : PA