The star-studded match between defending World Cup champions France and Germany was delayed briefly when an apparent Greenpeace protestor parachuted into the stadium before the game and was quickly detained by security.
After the match, Deschamps said: "We played a great game against very good opponents. I knew my players would be ready and we were up for the fight.
"We weren't far away from scoring the second goal that would have made us safe, but we didn't suffer that much in the second half. It was a strong match with quality and talent."
Karim Benzema, making his first appearance for France since the 2014 World Cup, played Adrien Rabiot into the area early in the second half, but the Juventus midfielder's shot caromed off the post as Germany clung to a one-goal deficit.
Another brilliant goal from France once again looked to have put the match away, but for the second time on the night Mbappe was called for offside after he had squared to Benzema for a close-range finish on the counter-attack.
Joachim Low's Germany, who had never lost their opening match at a Euro, battled to find an equaliser, but none came as France saw out the result to secure all three points to finish the day in second place of their group behind Portugal on goal difference.
The Germans, desperate to impress in front of a home crowd to make amends for their shock 2018 World Cup first-round exit, rarely broke through the France rearguard.
"We tried to play from the wings. We knew they were crowded in the middle and were tough on the one-on-ones. So we had to operate with crosses," said the German coach.
"Robin Gosens and Joshua Kimmich did enormous work in attack and we also had that chance with Serge Gnabry."
Meanwhile, the pairing of N'Golo Kante and Paul Pogba worked spectacularly for France, with Les Bleus having never lost a match -- 15 wins and four draws -- when the two in the starting lineup together.
UEFA weighed in on the parachuting pitch invader following the match, saying "several people" are being treated in the hospital for injuries caused by debris that fell, with the European governing body adding that, "law authorities will take the necessary action" for what it called a "reckless and dangerous" act.
Next up for France is a match against Hungary on Saturday, while Germany play Portugal in another juicy Group F match on the same day.
"It was a brutally intense game and we fought until the end," Low said postmatch. "An own goal decided it. I cannot blame the team because we fought a lot. We maybe lacked a bit of punching power in and around the box.
"Tomorrow we have to process things but look ahead. We lost, we are disappointed, but we still have two games and we can fix it."