Pilkington, a £2million summer signing from Huddersfield, opened the scoring after just 50 seconds before Russell Martin doubled the Canaries' lead soon after with a 12-yard header.
Danny Graham drilled home to pull one back for Swansea, but the Welshmen never looked like equalising and Pilkington sealed the win for the home side just after the hour when he swivelled neatly in the box before firing home from close range.
Paul Lambert's side were installed as favourites for the drop after their promotion, but they now sit ninth in the league after picking up 11 points from their opening eight games.
Swansea displayed little of the fight that saw them beat Stoke two weeks ago and are still in search of their first top-flight win in 19 years.
It was clear from the first 10 minutes who was going to win the contest of the two recently-promoted teams.
Just four hours after Wales exited the rugby World Cup, Swansea were looking to restore a bit of happiness to their nation, but they made the worst possible start and went behind after just 50 seconds.
Canaries striker Steve Morison knocked down Elliott Bennett's cross at the back post to find Pilkington, who fired past Michel Vorm from close range with a crisp half-volley.
Morison's hulking figure and sheer pace caused Swansea problems all afternoon and Garry Monk was perhaps fortunate to receive just a booking when he pulled back the Wales striker as he broke through the away defence.
Swansea could not get into the game and they found themselves 2-0 down in the ninth minute when Martin headed home unmarked from 12 yards after getting on the end of David Fox's pinpoint cross.
The two-goal cushion lasted less than two minutes, however, as Swansea raced up the other end to pull one back with their first real chance of the game.
Scott Sinclair skillfully skipped between Bennett and Fox and, despite falling to the ground, stuck a leg out to hook the ball across the box to Graham, who powered home from six yards for his second in two games.
Swansea started to play more confidently and Angel Rangel came close to equalising with a powerful 30-yard shot that flew just over.
Morison then battled his way past three Swansea defenders before laying the ball off to Pilkington, but he hammered in to the side netting from an acute angle.
Vorm then athletically tipped wide a Morison shot as the big striker kept bullying the away defence.
Swansea ended the first half the stronger side and they thought they should have had a penalty when Nathan Dyer went down after what looked to be a legitimate challenge from Bradley Johnson, but the referee waved played on.
Bennett beat his marker down the right in first-half injury time and floated over a speculative cross that landed on top of the crossbar and bounced out for a goal-kick.
The start of the second half was nowhere near as entertaining as the opening period of the first, with both sides playing scrappily in midfield and wasting chances up front.
Pilkington went for glory 35 yards out after some good work by Kyle Naughton and Wes Hoolahan but ploughed his shot in to the stand.
Manager Brendan Rodgers, unhappy with Swansea's start to the second half, brought on Stephen Dobbie for Wayne Routledge.
His hopes of a revival were quickly dashed, though, as slack marking allowed Norwich to score their third.
The ball fell to Johnson in the box after Swansea failed to clear a Canaries corner and he laid a calm pass in to the path of Pilkington, who cleverly swivelled and poked home from eight yards.
Morison's afternoon came to a premature end 25 minutes from time when he came off with what looked like an injury and was replaced by Grant Holt.
Rodgers brought on Leroy Lita for Dyer soon after and Vorm then saved well to deny Johnson from 25 yards.
Vorm had to be at his best moments later to keep out Holt's powerful 25-yard piledriver.
Dobbie wasted a chance to put Swansea back in the game when he miscued his shot after Graham had beaten two Norwich defenders to sneak into the box.
Lita had an opportunity to pull one back for the visitors in the last minute but his poor header flew at least five yards wide and Swansea went home to a despondent Wales with nothing.