The Chelsea striker scored twice in a friendly win over Ukraine, taking his tally to 42 international goals and passing France legend Michel Platini.
Arsenal legend Henry is the all-time record holder with 51 goals, but Giroud says it won't be easy to catch.
"Huge pride, it's a great honour, I'm very happy that it turned out that way," Giroud said.
"It was going to be emotional. I'm very happy for the team, we showed great things."