The 2008 finalists' challenge got off to a winning start in Lviv, courtesy of Mario Gomez's header 12 minutes from time.
Low admitted that it was a tough game for his side and said that they must show more creativity in attack.
"It was a really close game, but it has given us a good start," said Low. "We did well in defence; we worked hard, which wasn't always the case in our warm-up game.
"We have work to do up front with the tempo. It was there in phases, but we have to be more consistent."
The 52-year-old likened the European Championships to Formula One, stating the importance of a strong start.
"The European Championship is like a Formula 1 race without the warm-up," added Low. "You have to make a good start and get off the grid straight away; there are no easy games. You have to win your first match and we did."
Germany next face Holland in Kharkiv on Wednesday.