The Red Devils dispatched Eddie Howe's side 2-0 at Wembley to end a six-year trophy drought thanks to goals from Casemiro and Sven Botman's own goal.
Shaw was the architect of Manchester United's first goal while the victory was also the first trophy under Ten Hag.
The Old Trafford outfit are still alive in the Premier League, FA Cup and Europa League, which could provide further opportunities to land silverware this season.
Shaw believes the Carabao Cup can provide the platform for a period of sustained success under Ten Hag.
"It feels amazing. We said before that we want to create our own history here with the new team and the new manager, and I think [this] was the start of it," he told Sky Sports.
"I think the connection now is easy to see, we feel the love from the fans and we hope they feel the same from us.
"We wanted to start with a trophy and this is just the beginning for us."
Rashford echoed Shaw's thoughts, adding: "It's massive for us to be involved in these games, it's something that we've missed as a club."
"To come all this way and go on and win it, it's a massive feeling.
"Hopefully it pushes us to keep going now. The hunger is to have more moments like this. I've been in finals where we've won and where we've lost, and when you win it's massive for the club.
"It's part of our history at the club and we want to keep adding to that. I'm buzzing with the result.
"We want to be involved in a lot of games and staying in competitions. If we want to win things we have to keep pushing and keep going for everything."
