Gakpo, who played under Klopp at Liverpool, cancelled out Germany's 33rd-minute lead from Felix Nmecha, who plays at Klopp's previous former club, Borussia Dortmund.
Germany showed some high-pressing Klopp-football but came under a lot of pressure from the Dutch, who also had a high-profile new helmsman in Xavi Hernandez.
They had a let-off when a 13th-minute header from Oranje and Liverpool captain Virgil van Dijk was disallowed because of a foul by teammate Brian Brobbey on Germany keeper Marc-André ter Stegen.
But Gakpo then artistically volleyed home a deserved equaliser off a cross from debutant Ruben van Bommel, the son of former national team player and now Belgium coach Mark van Bommel.
It was a fresh start for both teams who had disappointed at the World Cup, going out in the round of 32 on the same day and on penalties, against Paraguay and Morocco, respectively.
It led to the resignation of coaches Julian Nagelsmann and Ronald Koeman and the subsequent appointment of Klopp and former Barcelona and Spain great Xavi.
Klopp gave three players a maiden cap in his rebuilding mission, with Freiburg right back Philipp Treu starting while teammate Yannik Engelhardt and Eintracht Frankfurt forward Younes Ebnoutalib came on.
Both sides continue on Sunday when Germany host Greece and the Dutch visit Serbia.
In their Scandinavian Group A4 clash, Norway's Erling Haaland struck twice, including a 74th-minute winner, to edge past Denmark 3-2 in Oslo.
The Danes powered back from a two-goal deficit and levelled on the hour after first-half goals from Oscar Bobb and Haaland had put the hosts 2-0 up. Norway were left with 10 men in stoppage time after David Moller Wolfe was sent off with a second booking.
Veteran Cristiano Ronaldo had a goal disallowed but Portugal still earned a 1-0 victory over Wales courtesy of Joao Felix's 22nd-minute long-range winner for a successful start for the defending Nations League champions' new coach Jorge Jesus.
Wales had several golden chances in the second half and hit the crossbar with Brennan Johnson before Portugal defender Ruben Dias made two key blocks to protect their lead.
Saied Abu Farchi scored in the 55th minute for Israel in their League B 3-1 defeat by hosts Austria, but the player was sent off with a second booking in the very next minute following his celebration where he simulated a gun with the corner flag.
Romano Schmid had given Austria a 43rd minute lead with a penalty and the hosts struck twice more late in the game to earn the three points.
There was drama in Belgrade as well where visitors Greece came from a goal down to snatch a 2-1 victory over Serbia thanks to Tasos Douvikas' winner deep in stoppage time.
France were nowhere on the teamsheet but French-born players scored every goal as Malta came from behind to beat Andorra 2-1 in their Nations League D clash.
Irvin Cardona and Teddy Teuma took turns as scorer and provider to secure Malta's victory after fellow French-born forward Guillaume Lopez had put the hosts ahead.
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