The 34-year-old hitman struck in both halves at London Road to earn Town a first away triumph of the campaign and end Posh's 100 per cent run.
Alexander broke the deadlock with a classy 22nd-minute volley from an acute angle after being picked out by Billy Clarke's exquisite cross from the left.
And the former Millwall and Leyton Orient frontman doubled the visitors' advantage with a powerful 65th-minute finish after the Posh defence had been masters of their own downfall.
Posh were left fuming at referee Darren Bond as their six-match winning spree in all competitions came to a miserable end.
Bond twice waved away strong penalty shouts in the closing stages as Britt Assombalonga hit the deck under challenges from Kyle McFadzean.
Tyrone Barnett and Gaby Zakuani both went close with headers as Posh failed to launch a comeback.
Crawley saw out the final few minutes with 10 men after Mike Jones earned a second caution for kicking the ball away.