The holders of the World Cup breezed past Poland to reach the quarter-finals.
The AC Milan forward scored his 52nd goal for France to become the country's all-time leading scorer.
The historic goal came on the stroke of half-time after Giroud controlled KylianMbappe's pass before converting low into the net.
After providing the assist for the first goal, the brilliant Mbappe turned on the style by scoring two wonderful strikes in the second half.
He hit a powerful drive into the roof of the net from 16 yards for his first and curled home a wonderful second in injury time to take the lead in the Goldenboot race with five goals.
The team was left frustrated until the opener and could have fallen behind.
In the first half, Poland's best opening was when the ball dropped for Zielinski in the center of the penalty area, but his strike was straight at Hugo Lloris.
Robert Lewandowski scored from the spot with the last kick of the game.
Next Saturday's quarter-finals will be played between England and France, with the winner going to the final.
When he left England for Milan, some people thought he was past his prime.
He helped the Rossoneri win the Serie A title for the first time in 11 years, scoring 14 goals in all competition.
He was desperately looking for that record goal against Poland and he was shaking his head whenever a chance came up.
He had a great chance when Ousmane Dembele cut back across goal, but he couldn't get enough contact on the ball.
After converting on 44 minutes, he was mobbed by his team-mates during the celebrations.
If captain Lloris lines up in the quarter-finals, he will break the record for most caps by a French player, currently held by Thuram.
Poland had to rely on Szczesny to keep them in the game at the half. Jules Kounde's shot from a narrow angle was the only goal of the game.
Szczesny was powerless to stop Mbappe from setting a new World Cup record by scoring seven goals before his 24th birthday. He scored both of his goals from the same spot in the penalty area to take his World Cup goals to nine in 11 matches.
He has scored at least one goal in every game he has played for France.
After Lloris moved off his line too quickly to save the first effort, the Polish forward stroked home a retaken 99th minute goal. The Poles haven't won against France in eight meetings.
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