United stunned the defending champions as they followed up a highly-contentious equaliser with a late winner to move within a point of their second-placed neighbours in the Premier League.
While arguments raged over the nature of Bruno Fernandes' leveller - awarded despite Marcus Rashford's run towards the ball from an offside position in the build-up - there was no denying the 2-1 win represented a strong statement from resurgent United.
Erik ten Hag's men have been transformed since they were thrashed 6-3 by City at the Etihad Stadium in October, losing just one of 19 games and winning their last nine in succession in all competitions.
Shaw, however, is not getting carried away with suggestions United could go on to overhaul Arsenal - opponents next Sunday - at the summit.
"It's too early to think about that," said the defender. "Right now as a team we have 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."
Shaw backed the officials in the row over Fernandes' 78th-minute strike.
City had looked on course for victory after substitute Jack Grealish's header on the hour.
That all changed after an offside Rashford raced onto a through-ball from Casemiro but stepped out of the way for the Portuguese to sweep home a shot.
As Rashford did not touch the ball he was deemed not to have interfered in play. City's protests were prolonged, continuing down the tunnel long after Rashford compounded their frustration with an 82nd-minute winner.
Shaw, who enjoyed another successful run-out at centre-back, said: "Of course you can understand where they are coming from - if I'm on the other team we're all doing the same - but I think Rashy was clever to know that Bruno was there and leave the ball. For me it was a goal."
City defender Manuel Akanji, by contrast, branded the decision to award the goal "a joke" but knows the team must quickly regroup after this latest setback, which followed Wednesday's Carabao Cup defeat at Southampton.
The Switzerland international said: "I hope we can improve in the next game. It was a different performance from Southampton - that wasn't a good game at all and this was a lot better - but still we lost twice. We really need to win on Thursday against Tottenham."