Longstaff marked a memorable Premier League debut with a wonderful strike in front of the Gallowgate End as the Magpies registered their first win of the season at St. James' Park, moving out of the relegation zone.
And Carroll, who was the first to congratulate Longstaff after racing towards a jubilant home crowd, revealed his pride for the Newcastle-born teenager after an impressive performance against the Red Devils.
"We're buzzing!" Carroll told nufc.co.uk. "I think we deserved it - we were the better team throughout. You couldn't write Matty's goal. It's his first Premier League game alongside his brother against Man United at home and he's scored the winner. You can't ask for any better and I'm absolutely delighted for him along with the rest of the lads.
"The lads are buzzing for him. Matty's a great lad who has worked hard. He's deserved his start today and kept his head down. He's grafted since I've been here and he deserved his chance and taken it with both hands."
After a heavy defeat against Leicester City a week ago, Carroll was pleased with the quick reaction from his squad as they earned their first league triumph since August - a 1-0 away win against Tottenham Hotspur.
With the international break looming, United will now wait a fortnight before their next top flight fixture - an away trip to Chelsea on Saturday, 19th October - and the striker believes the win over Ole Gunnar Solksjær's side was fully deserved following intensive work on the training pitch.
"I think it just shows what we have got in the squad," he added. "It was terrible last week and we've bounced straight back. We've worked very hard all week in training, and it has shown today as we dominated the game.
"It's important to show what we are about after last week. When you play a top team like (Manchester) United, it's always going to be a tough game but when you haven't won at home and you're struggling, like last week, I think it's a great win all round.
"The morale in the dressing room right now is great and we are buzzing heading into the international break. It's nice that we can celebrate for a couple of weeks and it eases the pressure on us. It's good going into the international break with a win rather than a defeat!"
