After Thaisa had cancelled out Valerie Gauvin’s 52nd-minute opener just after the hour mark, Henry settled a tense tie at Stade Oceane in Le Havre in the first minute of the second period of extra time.
The Lyon midfielder stole in front of covering defender Monica to sidefoot home from Amel Majri’s beautifully weighted right-wing free-kick to set up a last-eight clash with Spain or three-times champions the United States.
Following a cagey opening 22 minutes, France believed they had taken the lead only for VAR to again intervene in this tournament.
Kadidiatou Diani superbly turned Tamires near the corner flag on the right before whipping in an outswinging delivery to the edge of the six-yard box. Coming off her line, Brazil goalkeeper Barbara could only push the ball on to the shoulder of the incoming Gauvin, a collision that resulted in both players requiring treatment.
Only after a couple of minutes was a decision taken for the apparent goal to be reviewed, with Canadian referee Marie-Soleil Beaudoin deciding Barbara had been fouled.
It meant France ended the half without a shot on target, with Brazil’s best effort an acutely angled shot from Cristiane that was beaten away by the right foot of Sarah Bouhaddi two minutes from the break.
There was no doubt about France’s go-ahead goal in the 52nd minute as Diani again teed up Gauvin with a low, piercing ball into the six-yard box. On this occasion Barbara failed to intercept, allowing the Montpellier striker to stab home her seventh goal in her last nine internationals.
Brazil responded but Bouhaddi tipped on to the crossbar a 55th-minute looping header from Cristiane, who had met a deep free-kick from Marta.
Eight minutes later Brazil levelled, courtesy of VAR, as Debinha was initially flagged offside in receiving a through ball down the left. The move culminated in her delivering a cross that was cut out by Wendie Renard but straight into the path of Thaisa to firmly drive home from 12 yards.
Despite the offside flag, a quick consultation with VAR from Beaudoin ruled Debinha onside, giving Thaisa her fifth goal in her 65th appearance.
At 1-1 after 90 minutes, the game then swung within two minutes either side of the extra-time interval as just before the break Mbock Bathy Nka superbly cleared off the line a goal-bound shot from Debinha following a powerful run into the area.
Then in the minute after, Henry struck the winner to the delight of the partisan crowd.