A delighted Pompey boss Harry Redknapp said: "We pushed on after a bad start today and got the result we needed."
Pompey went a goal behind when Lewin Nyatanga struck after just five minutes and Redknapp added: "Benjani got three great goals but it was the crowd that lifted us."
The manager, who turned down Newcastle's mouth-watering offer to join them in place of sacked manager Sam Allardyce last week, added: "The fans could have got frustrated and moaned and turned against the team after so long without a goal but they don't do that here.
"The crowd's great, always has been for me, and that's one of the reasons I'm happy here. I've said Kevin Keegan is the right appointment for Newcastle - certainly their fans - and I wish him luck."
Benjani, who now has 12 goals this season, finally levelled with a shot on the turn on 39 minutes and then put Pompey ahead with a header four minutes later.
A sweeping move in combination with Pedro Mendes brought his third eight minutes into the second half.
Derby boss Paul Jewell saw his side extend their run without a win to 18 games and admitted: "It's the same old story I suppose. We gave them goals after taking the lead.
"But I love the job and I'm determined to bring some success to the people who keep coming to watch us. We had 1'200 fans follow us toady and they never stop cheering.
"Our problem was that we defended too deep and let too many crosses come into the box."