With the match looking set for extra time, Watkins received a pass from fellow sub Cole Palmer and fired a sweet strike into the far corner.
Xavi Simons had brilliantly put the Dutch ahead after seven minutes, with England equalising 11 minutes later with a Harry Kane penalty.
Chances came and went in an all action first half, things settled in the second before the late drama that sends England to play Spain in Berlin on Sunday.
It was a frenetic start as Simons robbed Declan Rice and slammed a screamer into top corner, and though the Orange Wall behind the goal erupted, it proved something of a false dawn.
England levelled when Kane was caught by the studs of Denzel Dumfries as he shot, earning a VAR-reviewed penalty that the England striker drilled into the bottom corner.
They were desperately close to a second five minutes later when Phil Foden brilliantly skipped through a series of tackles almost to the line and when he finally dug out a shot Dumfries was on hand to clear it off the line.
The Dutch centre back was in the spotlight again soon after when he rose magnificently for a corner but thumped his header against the bar, only for England to respond with a fantastic curling shot by Foden that hit the post with the keeper beaten.
Foden, Jude Bellingham and Bukayo Saka were playing with the swagger that they showed all season for their clubs while the incredible 19-year-old Kobbie Mainoo was running midfield like a veteran.
It was a scarcely believable transformation as the team who barely mustered a shot against Slovenia, Slovakia and Switzerland sent in crosses high and low, attempted probing through balls and even a couple of backheels in the box while forcing Bart Verbruggen into three sprawling saves.
The Dutch threw on striker Wout Weghorst for the second half while England brought on Luke Shaw but England seemed to lose their energy, playing safely.
For fans who saw the same pattern in the 2018 World Cup semi-final against Croatia and Euro 2020 final against Italy, where England were forced back after early dominance, it was a worry, but this time coach Gareth Southgate took dramatic action by removing Foden and Kane, bringing on Ollie Watkins and Cole Palmer after 80 minutes.
Saka did turn in a low Kyle Walker cross in a rare attack but it was ruled out for offside. However, Palmer, who has looked dangerous every time he has come on in the tournament, threaded a pass into the path of Watkins to cue bedlam.
It was heartbreak for the Dutch, who have now lost four semi-finals since winning the title in 1988.