Schweinsteiger came off the bench and scored within two minutes of his introduction in stoppage time to seal Germany's 2-0 Group C win over Ukraine in Lille on Sunday.
The 31-year-old, who came into the tournament with his fitness in question following an injury-interrupted debut season at Manchester United, converted Mesut Ozil's cross for his first international goal since 2011.
Speaking afterwards, Schweinsteiger revelled in his drought-ending strike, while dispelling fitness concerns.
"Everything's fine. My injury has healed, I feel good," Schweinsteiger said via SID.
"It was a long way for a celebration, I am a little bit out of breath.
"It was incredible. You could have only wished for such a thing to happen."
Team-mate Shkodran Mustafi got the ball rolling at the Stade Pierre-Mauroy, the centre-back heading in the opener midway through the first half.
Starting in place of the unfit Mats Hummels, Mustafi timed his run to perfection and placed a header into the top-left corner.
While Germany opened their Euro 2016 campaign with a win, Mustafi lamented the openness of the clash against Ukraine, who saw Yevhen Konoplyanka's effort cleared off the line by Jerome Boateng in the first half.
"It was important to start with a win, even without conceding a goal. It sometimes was an open match, an English match," said Mustafi.
"It was end-to-end stuff, something we did not want to happen."
Germany midfielder Sami Khedira added: "Well it wouldn't be right if everything was perfect in the first game. I think when our opponents analyse the match they'll see our weakness, and that was when we lost the ball - when we did, we should have reacted quicker.
"Getting the ball from the defenders to the strikers wasn't good but we got better in the second half."