Toni Kroos and Bastian Schweinsteiger scored a goal each for the Germans who are also chasing a record fourth European trophy.
The Germans did not look as stable at the back in the early exchanges of the match as Jerome Boateng was forced to clear off the line in the early minutes.
Goalkeeper Manuel Neur was also badly exposed at times and had to make good some saves. Ukraine could easily have scored late in the match as Yevhen Konoplyanka threatened a number of times.
But Ukraine’s attempts were ended when Schweinsteiger came off the bench in stoppage time. Despite his apparent lack of match fitness, the Manchester United midfielder was still able to score with a volley after the Germans had bewildered their opponents with a swift counter-attack.
Schweinsteiger who has not played since late March due to a knee injury will hope this goal can inspire him in the remaining matches of the tournament.
German are now top of Group ‘C’ and will face Poland on Thursday. Northern Ireland’s hopes of winning their first match at the European Championships ended in defeat after they went down 1-0 to Poland in a match played in Nice on Sunday.
Arkadiusz Milik scored the lone goal for Poland after slotting home a well-executed cross from Jakub Blaszczykowski early in the second half.
Despite dominating in the first half, the Poles simply could not get through the tough Northern Ireland backline. The only thing Northern Ireland got right in this match was to defend well as they were poor in creating chances and must do better against Ukraine on Thursday.
Turkey went down 1-0 to Croatia who will thank Luka Modric for scoring the only goal of the match in the 44th minute.
Croatia looked dominant in both halves coming into the match with a determination to pick all three points.
The winners were denied early goals when Ivan Perisic and Darijo Srna struck the woodwork. Turkey goalkeeper Volkan Babacan was a busy man for a good part of the tie to deny the Croats but in the end, he was beaten by Modric.
On Saturday, hosts France got off to a positive started after winning 2-1 against Romania while England was denied victory when Russia scored late to end the match 1-1. Albania and Switzerland also drew 1-1.