Luke Shaw replaced Kieran Trippier at the break, when Ronald Koeman turned to towering striker Wout Weghorst in place of Donyell Malen as he sought to change the dynamics.
The second period began with long spells of England possession without purpose, with the Dutch failing to click when they tried to threaten.
As Southgate’s side faded, the orange hordes found their voice and Pickford prevented them from celebrating in the 65th minute by showing quick reactions when Virgil van Dijk met a fine free-kick.
Kyle Walker made a timely challenge on Cody Gakpo, with Simons then striking an effort into the floor and the goalkeeper's grasp.
England did not so much as muster a second-half shot in response but thought they had scored in the 79th minute.
FULL MATCH DETAILS
Walker ran behind and Saka fired home his cutback, only for the flag to be raised for offside – a decision quickly ratified by those in the VAR booth.
Foden and Kane were replaced by Palmer and Watkins after that disallowed goal – an eye-catching double change that proved a masterstroke.
As fans and players braced themselves for another extra-time period, Palmer sent a low ball through for Watkins.
The angle looked too tight but the confident striker turned before Stefan De Vrij could react, lashing a low effort past Verbruggen in front of the stunned Dutch fanbase.
Koeman's men could not conjure up a leveller and the full-time whistle sparked bedlam as England progressed to Berlin.
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