The 25-year-old playmaker joined from Chelsea in a £37.1 million deal and made a winning debut against Cardiff City, assisting the second goal in a 2-0 victory.
Jones believes Mata's introduction has given a huge boost to the reigning champions, who have stumbled in defence of their title, and is confident the Spain international can have a huge impact at the club.
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“We want to impress and show him that we are what we are, that we are one of the best teams to be at. He came in against Cardiff. He has done well and I am sure he will be a huge asset and fantastic player for Manchester United, hopefully for many years to come.
“He sees the pass that sometimes no-one else sees. If he continues to make them, I am sure we will soon be climbing the table.”
Wayne Rooney and Robin van Persie both returned to action in Tuesday's clash with Cardiff, the latter opening the scoring, but Jones refused to blame United's problems on the strikeforce's injury absence.
“Of course [it has been a factor], but it is no excuse,” added the England international.
“When you do miss players of that stature you need to find it elsewhere. I think we have done it in certain areas but not in the areas we would like. We are just delighted to get them back and firing again.”