The Gills, who had handed a debut to striker Darius Henderson following his move from Reading, came close to snatching an early lead when Mamady Sidibe sprinted through the defence in the opening minute and forced keeper Patrick Kenny to save with his feet.
Injuries for United meant a place for Wayne Allison alongside Peschisolido up front and the latter soon found himself in space, but despite seeing off two defenders the final shot was poor and Gills keeper Nico Vaesen was able to deal with it comfortably.
United were the first to find another opening midway through the half when a cross from the left by Michael Tonge found Allison in the area. But the striker was unable to control the ball and wasted his opportunity by ballooning a shot over the crossbar.
Gillingham then enjoyed chances of their own when both Henderson and Danny Spiller came close to netting the opener.
In the 38th minute a Blades' attack saw Allison nod the ball to Peschisolido, whose left-footed shot beat Vaesen to give the visitors the lead.
With the goal advantage, Neil Warnock's men were the first to settle after the interval when an early throw in from Phil Jagielka found Allison in the area but the striker was unable to find a way past Vaesen.
The opening signalled a spell of further pressure on Gillingham who found themselves two goals down after 58 minutes.
Jagielka's cross from the left found Tonge unmarked in the area, who headed the ball across to Peschisolido and the striker made no mistake with a header of his own.
With confidence high, the Blades were now pressing for a third and Jack Lester came agonisingly close six minutes later when his shot came back off the crossbar.
The travelling fans did not have long to wait for another goal though as a minute later Peschisolido completed his hat-trick with Allison again providing the header on for his partner.
With the points already heading out of Priestfield Stadium, the Blades should have added a fourth to their tally in the last minute. But Lester, with only Vaesen to beat, failed to lift his shot over the keeper.