Lavreysen was pushed hard in the best-of-three final but the result appeared inevitable as he underlined why he is regarded as almost unbeatable in the power events.
The 27-year-old won both legs to add the sprint gold to the team sprint he won earlier in the week.
As he slowed down the hundreds of Dutch fans in the velodrome chanted "Harrie, Harrie" before he stopped on the banked track and lifted his bike above his head to salute them.
Lavreysen, who has 13 world titles across the sprint disciplines, will now go for a hat-trick of golds at this Games when he competes in the keirin event.
For Richardson, it was his first individual Olympic medal and adds to the bronze he took in the team sprint this week.
Before that, Italy won the women's Madison event, with Chiara Consonni and Vittoria Guazzini producing a stunning ride at the Velodrome.
Britain's world champions Elinor Barker and Neah Evans surged to the silver medal by winning the last sprint, while the Netherlands duo of Maike van der Duin and Lisa van Belle, who had looked on course for victory, bagged the bronze.
The 120-lap relay-like event in which teammates take it in turns to race while the others rest, with changeovers made using hand-slings, always produces drama and this was no exception.
Huge roars erupted around the velodrome as France took one of the first sprints to earn points but the Dutch acceleration caught everyone out as they lapped the field, a move worth 20 points in itself.
Italy then made a lap late on with an audacious attack which effectively earned them gold and left the others scrapping out for the other medals.
Barker powered around the last two laps to earn Britain the 10 points for the final sprint to boost them to silver.
