How did letting that one bank fail cause a world wide financial crisis?
Many international banks had interbank loans with Lehman. So when they went down, it hurt the liquidity of those banks.
The Lehman crisis hurt the whole American banking system in this way, the other banks had loans and securities in Lehman.
So when Lehman went down, the US market froze up, and also spread into other sectors.
Best thing would've been if the government would've moved in and completely took over Lehman for a brief time and sold off the shares after a few years when the liquidity in the bank was taken care of.
This is how the banks in Sweden were saved in the early 90's. UK has done the same for some of their biggest banks, which has prevented a major economic crisis in UK so far.