The Spain international made the £37.1 million switch to David Moyes' side from rivals Chelsea on Saturday, and made an immediate impact with a fine display in the 2-0 victory over Cardiff City at Old Trafford on Tuesday.
Mata was not on the scoresheet, but United took all three points as Robin van Persie marked his return from injury with a close-range header to open the scoring, while Ashley Young's solo effort confirmed the points on 59 minutes.
The 25-year-old also got to link up with England international Wayne Rooney, who returned from a five-match absence with a foot issue for the final 30 minutes.
And Mata revelled in making his United bow.
"It's been unbelievable, playing in this stadium, we won and created chances," he told BBC Sport. "Three points was important for the team and for me, I'm very happy.
"I was very excited, it's been unbelievable. I'm happy with the team, they have welcomed me. I'm looking forward to playing more."
When asked whether United, who currently sit seventh, can reach the Champions League, he added: "We will try until the end.
"This is a team that never gives up. Every game is like a final for us."
Mata was also thrilled to get game time with Van Persie and Rooney and believes the duo will flourish for the rest of the campaign.
"We are talking about two of the best strikers in the world," he continued. "Robin and Wayne are unbelievable. Hopefully they will score lots before the end of the season."
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