The 23-year-old registered his first senior goal in his 134th appearance in all competitions as the Red Devils beat Leicester City 2-1 in Friday evening's Premier League kickoff.
Following a tough couple of seasons under Jose Mourinho, when having his performances publicly criticised by his manager, the seven-cap England international acknowledges that he cannot afford to get too carried away.
"It was my first goal and that is all that was important, it doesn't matter how I scored," he told Sky Sports News. "I was meant to come inside and I managed to get in front of him and put it in. It's the best feeling I have had in my career.
"I have worked really hard over the pre-season and I had a lot of time to reflect, and work on what I needed to. Even tonight, I know I can be better and that is what I want to keep showing the manager.
"I don't want to speak too soon after one game but I want to be up there with the best left-backs in the world. I have the foundations to do that here with the manager and my teammates at Manchester United.
"I want to play for England and I want to be involved again. I don't think I was playing as well as I could with my attacking capabilities last season, and that's what I'm known for."
Shaw was wearing the United shirt for a 67th time, matching the number of appearances he made for former club Southampton.