Deschamps' France lost 5-4 on penalties to Switzerland after Kylian Mbappe's spot-kick was saved by Yann Sommer, squandering a 3-1 lead with 15 minutes of regulation time remaining in Bucharest on Tuesday (AEST).
France failed to reach the quarter-final stage of a major tournament (European Champion and World Cup) for the first time since the 2010 World Cup following the shoot-out against Switzerland after the dramatic 3-3 draw at the end of extra time.
Deschamps is contracted until 2022 and has been in charge of Les Bleus since 2012, lifting the World Cup in 2018 and finishing European Championship runners-up in 2016.
France's premature Euro 2020 exit led to questions about the former France international's future as national team boss amid links with former Real Madrid boss and countryman Zinedine Zidane.
"That is not the question," Deschamps said after the defeat. "There is a unity and solidarity in this squad.
"I am responsible when things go badly - I am with them, they are with me. We will need to time to manage this, it hurts tonight."
France superstar Mbappe had his spot-kick saved in the decisive shoot-out moment by Sommer as Switzerland reached the quarter-finals of a major tournament for the first time since the 1954 World Cup.
Paris Saint-Germain's Mbappe had more shots (14) without scoring than any other player at Euro 2020.
Deschamps refused to blame Mbappe, adding: "Nobody can be annoyed with him.
"When you take the responsibility, it can happen. He is obviously very affected by it."
Deschamps said France showed weakness by allowing Switzerland's two late goals which forced extra-time.
France had fought back from a first-half deficit after Haris Seferovic's 15th-minute opener, with three second-half goals in 18 minutes, initially a Karim Benzema double before Paul Pogba's stunning strike.
Switzerland pulled a goal back with Seferovic's close-range header before Mario Gavranovic found space to level in the 90th minute to force extra time.
France has been eliminated in its last three games in which they played extra time in major tournaments (Euros and World Cup), as many as in their first 11.
"It is always complicated to explain," Deschamps said. "We failed with our first half, we did what was needed to turn it around in the second half.
"Usually our strength is being solid, we showed weakness that allowed Switzerland back in. This is hard, it hurts, we did everything we could for this to end differently. That's football.
"This tournament ends for us today. There is no magic formula."