Nicola White’s last-quarter strike levelled the match at 3-3 and the game went to penalties.
Helen Richardson-Walsh converted a penalty flick after Georgie Twigg was fouled and Webb kept her cool to net the decisive effort after Hinch had time and again denied the Dutch.
It meant Britain, London 2012 bronze medallists, were champions as Holland’s bid for a third successive Olympic gold came to an end.
Lily Owsley, Crista Cullen and White scored in normal time, when Hinch continually kept Britain in the contest, while Kitty van Male scored twice and Maartje Paumen once for world champions Holland.
Britain’s task was a tough one. None of the players in the current Dutch squad had ever lost a match at the Olympics.
Holland, winners in 2008 and 2012, last lost in the 2004 final to Germany and had one defeat in 12 prior meetings with Britain at major tournaments. That was in the 1989 Champions Trophy.
Bronze in 1992 and four years ago was the previous best performance by a British women’s team, but they wanted the nation to stop talking about the men’s win in 1988 in Seoul.
Holland squeezed through in a shootout against Germany, who won bronze ahead of New Zealand, and were determined to make a good start.
Britain were indebted to goalkeeper Hinch for saving Paumen’s penalty flick after Sam Quek’s rash challenge on Laurien Leurink.
Kate Richardson-Walsh blocked Naomi van As’ shot as Holland threatened once more before Sophie Bray conjured up a goal for Owsley with a juggling run and shot. The ball rebounded for Owsley to sweep in her fourth of the tournament.
Holland took the direct route to equalise as Lidewij Welten held off Kate Richardson-Walsh and fed Van Male, whose reverse-stick strike found the net.
Cullen was sin-binned for two minutes and Hinch made a double save following a penalty corner to keep Britain level.
Van As hit the bar with another reverse strike.
Two more Dutch penalty corners followed and Paumen stroked in from the second of them. It was her 195th goal on her 235th appearance.
But Cullen found a gaping hole on the Dutch left and swept the ball in.
Holland penned Britain deep in their own half in the third quarter, forcing a succession of penalty corners.
The Dutch breakthrough came when, instead of shooting, Paumen returned the ball to Van Male to tuck in from close range and restore the lead.
England were 2-0 down at the end of the third quarter in the European Championship final last year against Holland and responded to draw 2-2 and win on penalty flicks. So Britain knew they were not out of it with 15 minutes remaining.
Britain pressed forwards and White bundled in following a penalty corner to equalise.
And the destination of the gold medal was determined by a penalty shootout.
Helen Richardson-Walsh and Alex Danson were denied, but Hinch twice kept the Dutch at bay to keep Britain level.
Twigg was fouled by Dutch goalkeeper Joyce Sombroek, so the penalty was upgraded to a flick and Helen Richardson-Walsh scored.
Hinch saved again, this time from Leurink, but Laura Unsworth blazed her effort over.
Margot van Geffen beat Hinch, but hit the post, meaning if Hollie Webb scored Britain would be champions. And she duly delivered.