Back-to-back wins over Mark Hughes' men and Real Sociedad in midweek have gone some way to easing the pressure on Moyes, who could ill afford any more slip-ups after United's sluggish start to the new season.
And despite last-gasp goals from Wayne Rooney and Hernandez dragging United out of another hole, Carrick acknowledges much improvement is still required if they are to claw their way back into the title race.
Carrick said: "We've been saying for a number of weeks when we've had a good result we have to kick on and get back to our normal selves.
"We've still got a way to go, we're not kidding ourselves. We've had a couple of good results but by no means are we at our best - we're not in bad shape but there's more to come.
"There's no point looking at the table at the moment. Obviously we're aware of the situation but there's no point going over it in your head thinking 'what if', we've just got to concentrate on getting the wins and see where that takes us."
United could hardly have wished for more pliant opponents as they sought to get their season back on track, as Hughes' men travelled to Old Trafford having scored just four goals from their first eight Barclays Premier League games.
But it took just two minutes for the visitors to make a mockery of those statistics as their excellent left-back Erik Pieters took advantage of space on the left flank to cross for Crouch, who bundled past David De Gea at the second attempt.
The visitors attacked with pace and purpose and could have made it worse for United, Crouch spooning over the bar after 11 minutes then Jonathan Walters firing a strong shot from eight yards which was brilliantly saved by De Gea.
Even when United did wrest their way back into the match in the 43rd minute, Robin van Persie bundling home after Asmir Begovic could only palm a Rooney effort into his path, Stoke regained their lead almost immediately from Arnautovic's superb set-play.
Stoke had more chances in a slower-tempo second half before the introduction of Adnan Januzaj and Hernandez inspired United to grab their 78th minute equaliser, Rooney providing the deftest of flicks to a Van Persie corner.
And there was a mighty roar of relief two minutes later when Hernandez headed home a cross from Patrice Evra from the opposite flank to ensure United would not lose any more ground on the leaders.
Carrick added: "It's the character we have - it was a vital game to win. We didn't really have a choice, we had to find a way and we kept going and believing and ended up finding that win."
Defeat was hard to take for the visitors but captain Ryan Shawcross believes it was the type of performance which can stand Hughes' men in good stead for the rest of the season.
Shawcross said: "Obviously we're disappointed but it's great that we came to a big team and put in a performance like that.
"They (United) have struggled so far this season so of course we came here with more belief on the back of the poor results they've had. But they're still a great team and they showed that in coming back to get the victory.
"We've got a good run of games now, five or six that we should take points from. The gaffer has set us a target and we've got to achieve it. It's going to be hard but we can take some belief from a performance like that."
- Sports & Recreation
- Wayne Rooney
- Peter Crouch
- Marko Arnautovic