The experinced midfielder got on the end of a perfectly weighted Ozil cross to double his side lead during Germany’s 2-0 win over the weekend.
Ozil has highlighted Schweinsteiger’s importance in Germany’s Euro 2016 campaign.
“I think it was important for him,” Ozil told reporters after the game.
“He was injured for a long time and with that goal he’ll probably gain a lot more self-confidence.
“But the coach has also often said it — we need every player. It’s a long tournament and we need everyone to be fit.
“So we’re happy that Bastian got a goal because he’s an important player for us.”
Schweinsteiger has struggled in his first season at Manchester United. The midfielder has been sidelined with injuries and there were doubts about whether the 31-year-old would be fit enough to feature in Euro 2016.
Ozil was one of the stars of the show for Germany against Ukraine. The Arsenal playmaker maintained a passing accuracy of 88%, made 49 passes and created three chances.
The 27-year-old has enjoyed an impressive season winning the Arsenal player of the season award and proving the highest number of assists in the league (19).