Les Bleus were well off the pace during the opening 45 minutes in Sarajevo, but the 2018 world champions ultimately sealed all three points thanks to Antoine Griezmann in the 60th minute.
The Barcelona forward met Adrien Rabiot's left-wing cross with a superb header that brought up his 35th international goal.
An eighth consecutive away win - the longest winning streak on the road in their history - moved them four points clear of second-placed Ukraine at the summit of Group D.
While accepting his side's first-half performance was not good enough, Deschamps was largely pleased with how his players battled through a third game in eight days.
"Our first-half display is below what we are capable of doing," he told M6.
"It was better in the second half. It is not easy. All matches are complicated. There are things that I have seen us do that I haven't seen in a while. The main thing is victory.
"We did more in the second half even if I take the opponent into account as well. I found that the start was quite difficult. I don't want to find excuses for the players. I am very demanding with them, even if the trips and the time differences are difficult.
"The three matches we will have in September will be against our three direct opponents. Nothing is done but we have put ourselves in the best place with seven points from three games."
Hugo Lloris, whose 123rd appearance moved him level with Thierry Henry as France's second-most capped player, attributed their ability to grind out a result to their robust mentality.
"It was a bit laborious, especially in the first half," the Tottenham goalkeeper said.
"We struggled with the use of the ball especially. They put us in danger on the counter-attacks and on the set pieces with the quality of [Miralem] Pjanic. But the state of mind is remarkable. Honestly, we played badly today, but we won with the state of mind.
"It's something you can build on. This is not the first time, but we must continue to move forward, to improve because there are important deadlines in June and July. After Antoine's goal, we managed the result.
"We could have scored a second goal, but we got the win and we are moving forward."
Le Graet has withdrawn from his duties pending a French government investigation into the governing body, with French sports minister Amelie Oudea-Castera calling for change at the top of the organisation.
That news came just three days after Le Graet caused controversy with his bizarre comments about Zidane in the aftermath of Deschamps signing a new four-year deal as France coach.
Zidane was linked with Deschamps' job following last month's World Cup final defeat to Argentina, but Le Graet said he would not "give a toss" if the former Real Madrid coach expressed an interest in leading the national team.
While Le Graet subsequently apologised, his comments were labelled as disrespectful by Kylian Mbappe and were also seen as distasteful by Deschamps.
"His words, as he acknowledged and admitted, were inappropriate," Deschamps said when asked about Le Graet on Wednesday. "I think it's a very good thing that he apologised to Zizou."
Deschamps then acknowledged constant speculation linking Zidane with his job had created a rivalry between the two coaches, who played together for Juventus and the national team.
"The sporting situation has led to a rivalry between the two of us. Even, for some, an opposition," he said of his relationship with Zidane.
"I will always have a lot of respect for him in relation to what we shared together in our first life as players, and for who he is, what he represents in football and French sports."
While Deschamps will lead France at a fourth World Cup in 2026 after penning his new deal, goalkeeper Hugo Lloris will not be present in the United States, Mexico and Canada after announcing his international retirement.
Deschamps was sorry to see his captain call time on his international career, saying: "I spoke at length with Hugo, I know him very well. I will not go into details.
"When a person is driven by a deep conviction, he does not make a decision on a whim. It was carefully thought out. I understand his decision, even if I regret it.
"He is someone who has been very important to the France team both on a human and a sporting level. We will miss him, but it is his decision, and we must say thank you for all he has done."
Lukaku and Immobile are two of the form front men in European football, and the duo have earned recognition following their latest goalscoring exploits.
Inter striker Lukaku grabbed both goals in a 2-0 away win at Udinese on Sunday.
And Immobile matched that haul as Lazio thrashed SPAL 5-1, the Italy international's brace taking his 2019-20 league tally to 25 already.
The duo are far from the only big names in this week's selection, however, with Roberto Firmino, Jadon Sancho and Hugo Lloris also among the new in-form cards.
Find the full squad below.
TEAM OF THE WEEK
GK: Hugo Lloris (Tottenham) - 89
CB:Matthijs de Ligt (Piemonte Calcio/Juventus) - 87
LB: Alex Telles (Porto) - 87
CB:Yerry Mina (Everton) - 84
CM: Daniel Parejo (Valencia) - 88
RM: Angel Di Maria (Paris Saint-Germain) - 88
CAM: Jadon Sancho (Borussia Dortmund) - 87
CM: Santi Cazorla (Villarreal) - 86
CF: Roberto Firmino (Liverpool) - 88
ST:Ciro Immobile (Lazio) - 90
ST: Romelu Lukaku (Inter) - 88
SUBSTITUTES
GK:Steve Mandanda (Marseille) - 83
CB:Kaan Ayhan (Fortuna Dusseldorf) - 81
RM:Robert Snodgrass (West Ham) - 81
ST:Francesco Caputo (Sassuolo) - 84
RW:Daniel Ginczek (Wolfsburg) - 82
ST:Kasper Dolberg (Nice) - 81
LW:Oussama Idrissi (AZ) - 81
RESERVES
CM:Alexandru Maxim (Gaziantep) - 79
CM: Alexandru Cicaldau (Universitatea Craiova) - 78
LM: Ahmet Engin (Duisburg) - 76
LM: Nathan Thomas (Carlisle United) - 74
LW: Said Benrahma (Brentford) - 80
Goalkeeper Lloris is Les Bleus' most-capped player and has captained the team since 2012.
The Tottenham man skippered his country at six major tournaments, including three World Cups.
Lloris was the man between the posts as France won their second world title in 2018, having lost in the final of their home Euro 2016 two years earlier.
With Lloris still in goal, France returned to the World Cup final at Qatar 2022, only to lose on penalties to Argentina.
Having taken time to announce a decision following that defeat, Lloris confirmed his retirement in an interview with L'Equipe on Monday.
"There comes a time when you have to know how to hand over," he said, before referencing deputy Mike Maignan.
"I don't want to own [the position]. I've always said and repeated that the France team doesn't belong to anyone, and we all have to make sure that this is the case, me first.
"I think that there remains a team ready to continue, and there is also a goalkeeper who is ready."
Maignan appeared set to push Lloris for the number one shirt in Qatar, only to join a long list of France stars who missed the tournament through injury.
Lloris kept only a single clean sheet at the finals, although that came in the semi-final against Morocco.
He retires with eight World Cup clean sheets, behind only countryman Fabien Barthez and Peter Shilton (both 10) since 1966.
"I'd rather go out at the top than wait for a decline or [face] too much competition," Lloris added. "There is also a family choice;I feel the need to spend more time with my wife and children."
That is the verdict of Paris Saint-Germain head coach Christophe Galtier, who steered clear of either recommending the role to Mbappe or putting any pressure on national coach Didier Deschamps.
At the age of 24, Mbappe has already won 66 caps for Les Bleus, scoring 36 goals in the process.
He won the World Cup in 2018 and scored a hat-trick against Argentina in the 2022 final but painfully finished on the losing team.
PSG striker Mbappe appears an obvious candidate, but he may not aspire to the captaincy, which Lloris held for 10 years.
Manchester United defender Raphael Varane is another strong contender for the role, according to reports in France, having served as vice-captain at the World Cup.
Galtier said on Tuesday: "I'm not the national team coach. I'm not going to get involved in the France captain discussion. Sometimes, the France captain is of a certain age with a certain number of caps.
"Kylian Mbappe has got many under his belt and many incredible performances. I haven't spoken to Kylian Mbappe about whether he will be the France captain or not.
"Of course, that is a huge responsibility. It is not me who will decide who the captain will be. Even without the armband, he is a leader."
Varane, who made his France debut in March 2013, made 93 appearances for Les Bleus.
The former Real Madrid centre-back, who joined Manchester United in 2021, played and started in all seven games of France's successful 2018 World Cup campaign.
Despite heading into last year's World Cup with an injury, Varane featured in six of France's matches in Qatar, including in the final against Argentina.
On Wednesday, Varane shared an Instagram post confirming his retirement from the international stage.
"Representing our great country for a decade has been one of the greatest honours of my life," Varane wrote.
"Every time I wore this very special blue jersey I felt immense pride. The duty to give it [my] all, play my heart out and win every time we went on the field. I've been contemplating this for several months and decided it's the right time for me to retire.
"As a child, I remember following France 98, this team, these players who made us experience indescribable emotions. I dreamed of being our heroes and 20 years later I lived one of the best experiences of my life.
"We brought home the World Cup!! I will never forget it. I still feel every single one of the emotions felt on that day, July 15, 2018. That was one of the most amazing and memorable moments of my life.
"This victory that we won all together would never have been possible without the support of [coach] Didier Deschamps, every member of the management and staff over the years, and my team-mates who have defended our shirt in every game we played.
"But more importantly, this victory would not have been possible without every single one of you supporting us along the way. Even after the defeat in the final last year, you welcomed us as heroes when we returned. To each and every one of you, THANK YOU a thousand times!
"I will miss these moments with you for sure, but the time has come for the new generation to take over. We have a bunch of talented young players who are ready to take over, who deserve a chance and who need you."
Varane, who captained his country 20 times, is the fourth member of France's 2018 World Cup squad to retire from international football, after Blaise Matuidi, Steve Mandanda and former captain Hugo Lloris.
Karim Benzema did not make that squad but was set to feature in Qatar until injury derailed his chances. The Ballon d'Or winner announced his retirement following the tournament.
Deschamps said: "Raphael called me a few days ago to explain to me that he wanted to put an end to his international career. He is an intelligent boy, who knows how to take the time to reflect, weigh the pros and cons before deciding.
"Without going into the details of our exchange, Raphael believes that he has reached the end of his adventure with the France team. I experienced a situation somewhat similar to his, I understand his arguments and respect his decision even if it may seem regrettable.
"Of course, I cannot turn this page without a certain emotion, given the ties we have forged since August 2012. Like Hugo Lloris, he will have been, barring injury, in all of my squads."
