Mbappe's 82nd-minute strike against Ukraine on Friday, set up by his Real Madrid teammate Aurelien Tchouameni following a swift counterattack, lifted him to 51 international goals, level with Henry in second place on France's all-time scoring list, behind Giroud's 57.
France opened their World Cup qualifying campaign with a commanding 2-0 victory over Ukraine, with Michael Olise scoring early before Mbappe sealed the result late on at the Wroclaw Stadium in Poland.
"Big up to Titi (Henry)! But now I want to surpass him! It's an honour to equal a player like Henry," Mbappe, 26, told TF1.
"Everyone knows what he means to us French people, even more so to strikers. He's someone who paved the way, and I have a lot of respect and admiration for him.
"Reaching this milestone so early is crazy, but I like it. I want to keep going and, above all, win games and titles."
Mbappe is firmly on track to surpass Giroud, who ended his international career after the conclusion of the 2024 European Championship, where France lost to eventual champions Spain in the semi-finals.
Since making his France debut in 2017, Mbappe has already enjoyed a glittering international career, including the 2018 World Cup triumph and the 2020-21 Nations League title.
"The record is getting closer, but it's not something I think about," the France captain said.
"I don't know if it's because I think I can beat it, or because I think there are more important things.
"But it's true that being the top scorer in the history of the French national team is no small feat."
France next host Iceland in Group D at Parc des Princes on Tuesday.