The former Preston North End striker was constantly jeered throughout the evening by the home supporters, but put all that behind him on 69 minutes to score the only goal.
The home side started brightly and created the first opportunity of the night when central defender Clarke Carlisle blazed a shot over the crossbar from an Alan Mahon corner .
Nugent should have put the visitors in front after 18 minutes when he latched on to a through ball from Pedro Mendes but Clarets goalkeeper Gabor Kiraly dived at the feet of the England striker
Mahon was again involved in the action shooting wide from 25 yards after a good knock-down by Kyle Lafferty.
An Mahon's corner just after the half-hour mark was met at the far post by Burnley skipper Steven Caldwell, whose header was blocked by Pompey keeper Jamie Ashdown at the expense of a corner.
A cross from the right by Benjani Mwaruwari found its way through to Matthew Taylor and his right-foot shot produced an outstanding save from Kiraly diving low to his right.
A further Mahon corner just prior to the half-time interval was met on the near-post by Ade Akinbiyi but he placed his header straight at Ashdown.
Pompey boss Harry Redknapp replaced Benjani with summer signing John Utaka at the half-time interval.
Seven minutes into the second half, left-back Djimi Traore crossed to the far post where Nugent headed back across goal but wide of the target.
Northern Ireland international Lafferty tested Ashdown from 20 yards after a neat pass from David Unsworth.
The only goal of the cup tie arrived on 69 minutes when Nugent latched on to a long ball from Mendes and tucked the ball into the far corner beyond Kiraly.
Five minutes later Nugent had an opportunity for his second after being played in by Traore but again shot wide of the target.
With 14 minutes remaining Burnley manager Steve Cotterill made a double substitution in an attempt to grab an equaliser replacing James O'Connor and Unsworth with Robbie Blake and Graham Alexander.
But although Burnley's new formation pressed hard, chances were few and far between with Ashdown fumbling a volley from Mahon and substitute Blake having a shot blocked by the Pompey rearguard.
But it was Nugent who was to have the last laugh much to the pleasure of his manager.