Substitute Ashley Harris struck in the 74th minute on Sunday to put Pompey ahead before Luke Rodgers and Izale McLeod completed the away side's victory - ending Town's 100% home form in front of a record 5,058 crowd at the Broadfield Stadium.
Crawley finished the match with just nine players after Josh Simpson and Hope Akpan were both sent off for two bookable offences.
Akpan had gone close for Crawley with a dipping right-footed volley from 20 yards before Gary Alexander almost steered home a teasing Nicky Adams cross as the Reds began brightly. But in the 41st minute referee Darren Deadman showed a red card to midfielder Simpson after two separate fouls on McLeod.
Team-mate Akpan was then given his marching orders in the 71st minute before Harris quickly tapped Pompey into the lead after a cross from Adam Webster.
Rodgers doubled the lead with a stunning 20-yard free-kick in the 83rd minute before McLeod rounded Crawley 'keeper Paul Jones to complete his side's victory two minutes later.