Targett's impressive low finish opened the scoring for the young Magpies on Tyneside before Remi Savage's header, courtesy of the left-back's teasing corner, doubled the hosts' lead.
The Bees halved the deficit in the closing stages of the second-half through Michael Olakigbe's close-range finish before Ben Parkinson's strike and Rodrigo Vilca's excellent volley sealed an emphatic victory for Ben Dawson's side.
After observing a minutes' silence in memory of former Newcastle winger Christian Atsu, who tragically lost his life in Turkey's devastating earthquakes, United's second-string almost got off to a flying start through Parkinson as the Durham-born forward, who scored his first goal at under-21 level against Leeds United in the Premier League 2 on Monday, saw his low shot blocked by Brentford goalkeeper Roco Rees before the follow-up was parried away by the Bees shot-stopper.
In the 12th minute, Newcastle took the lead through Targett. Following dangerous build-up play on the left flank, Michael Ndiweni produced a neat flick into the full-back's path who buried a low attempt into the bottom right-hand corner.
Seven minutes later, Newcastle extended their advantage through Savage as the 21-year-old centre-back rose high from Targett's inviting corner before heading the ball into the far right-hand corner.
Newcastle faced attacking pressure from the West London outfit before the half-time break as Bees winger Romeo Beckham, son of former England captain David, lashed an attempt over the crossbar.
Vilca, who spent the majority of 2022 on loan at Universitario de Deportes in his native country, went close to adding a third as the Peruvian midfielder fired a curling effort wide of the right-hand post.
Ndiweni, who signed his first professional contract with his boyhood club last summer, tested the Rees shortly after, with the Throckley-born striker beating the offside trap before electing to shoot from a tight angle, which was pushed away from goal.
In the 65th minute, Newcastle's lead was halved as Olakigbe netted for the visitors, latching onto a ball inside the penalty area before drilling beyond Newcastle goalkeeper Max Thompson.
Despite Brentford's effort to return to level terms, Newcastle regained their two-goal cushion in the 86th minute through Parkinson's tidy finish, with the ball landing into the 17-year-old's path before firing a half volley into the net from the edge of the penalty area.
Vilca added gloss to the scoreline in the 89th minute as the 23-year-old, who joined the Magpies in October 2020, directed an incredible volley from distance into the net to complete a commanding victory.
Newcastle United Under-21s: Max Thompson, Ciaran Thompson, Charlie Wiggett, Remi Savage (c) (Jordan Hackett 60), Matt Targett (Matt Bondswell 60), Nathan Carlyon, Lewis Miley (Jamie Miley 45), Ben Parkinson, Rodrigo Vilca, James Huntley (Amadou Diallo 45), Michael Ndiweni.
Subs not used: Will Brown.
