The Red Devils came from behind to beat neighbours Manchester City 2-1 at Old Trafford on Saturday.
Bruno Fernandes and Marcus Rashford were on target for the home side after substitute Jack Grealish had given Pep Guardiola's side the lead.
Manchester United have now won their last five Premier League games and are nine points behind leaders Arsenal.
Confidence is brewing in the Manchester United camp but Shaw says none of the club's players is getting carried away by their good form.
"It's too early to think about that [winning the league]," Shaw said. "Right now as a team we have to make sure we have the right consistency in performances and results.
"It's always about winning. If you play well or not you need to be winning, and recently we've been doing that. We've not been playing our best but we've still been winning games.
"We're still in four competitions. We're Manchester United, we have to aim high, and of course we want to win all four but whether that happens or not is down to us, it's down to the way we act, the way we rest and get ready for the next game.
"Now we have to forget about this game, forget about this win. We have a massive game on Wednesday against Crystal Palace."
