Former Bayern Munich boss Flick took over the reins from Joachim Low in May 2021 and the 57-year-old had overseen wins in each of his eight games in charge before the trip to Amsterdam.
It looked like he would make it nine on the spin when Thomas Muller emphatically slammed home his 43rd international goal before the break.
Louis van Gaal's side had the final say, though, with Bergwijn powering home the equaliser midway through the second half.
Germany started brightly and went close in the 11th minute when Leroy Sane drilled into the side netting from a tight angle.
They threatened again 10 minutes later when Timo Werner's header from David Raum's cross crashed back off the underside of Mark Flekken's crossbar.
Donyell Malen then clipped just wide of Manuel Neuer's right-hand post under pressure from Antonio Rudiger in what was a rare sight of goal in the opening period for the hosts.
Germany deservedly went ahead in first-half stoppage time, with Muller lashing a crisp left-footed strike past Flekken after Jamal Musiala's cross had been deflected into his path.
Die Mannschaft's failure to find a second goal was punished in the 68th minute when substitute Bergwijn thundered home from six yards following Denzel Dumfries' downwards header.
The Netherlands were denied the chance to score a winner soon after when referee Craig Pawson overturned his initial penalty decision for Kehler's foul on Depay after viewing the pitch-side monitor.