James, who completed a £15m move from Swansea City last month, was named in the starting XI for the 2-0 pre-season friendly win against Perth Glory.
The 21-year-old winger caught the eye and was named man of the match, despite being withdrawn at the break.
Manchester United boss Solskjaer told MUTV: “He is sharp and he started well.
“It was a good start for him and we’ll nurture him and he’ll be exciting to watch for many, many games I think.”
Marcus Rashford, who has attracted some criticism for his lack of goals, opened the scoring in the second half with a brilliant finish from a tight angle.
And Solskjaer hopes that is a sign of things to come from the England international.
“I’m very pleased with the goal he scored,” the Norwegian said. “It was instinctive. He swivels and finishes.
“It’s not one of those pretty goals, but it’s a goal that I’ve said that Marcus needs to add to his tally, because he scores wonder goals but when he can get five of these a season, hopefully he can get some like it in the league as well, he’ll add to his tally.”
James Garner, who came on as a late substitute for the injured Luke Shaw, banged home a terrific finish to double the score.
He has been identified as a hot prospect, and Solskjaer was pleased for the rising star.
“It’s great for James,” he said. “We’d just had a corner kick and I said to him: ‘get on the edge of the box and score with your first touch’, but I think he had two, so I wasn’t happy with that one!
“It was a fantastic strike, the boy can play and he’s enjoying tonight.”
Solskjaer was pleased with the comfortable success despite Perth Glory sitting deep, making it difficult.
“It was a game where we had the ball all the time more or less and we kept possession well,” he said.
“We played against a well-organised, disciplined team that didn’t really give us too many gaps. We created half-chances.
“We got 45 minutes of work for most of them, which was the main objective of today, and of course you always like to win games.”