The floodgates opened straight away at Carrow Road with Sargent and Greek winger Christos Tzolis, also making his full debut, each scoring twice.
The Canaries were 3-0 up by half-time with Championship side Bournemouth managing only one shot on target throughout the 90 minutes.
In some of the other day's matches, Rafa Benitez's Everton were down to 10 men after 59 minutes at Championship side Huddersfield Town when forward Moise Kean was shown a straight red with the score at 1-1, but Andros Townsend grabbed a 79th minute winner for the visitors.
The goal was new signing Townsend's first for the Toffees and came after Tom Lee's header on the stroke of half-time had cancelled out Alex Iwobi's 26th minute opener for Everton.
Kean had the ball in the net but was ruled offside and then saw red after an off the ball incident.
Brentford, flying high on their return to the Premier League this year, were a goal down at home to fourth-tier Forest Green Rovers but ran out 3-1 winners with three second-half strikes.
Forest Green played with 10 men for the last 10 minutes after Ebou Adams was sent off.
Patrick Vieira's Crystal Palace were the sole Premier League casualties of the evening, losing 1-0 in their all top-flight clash at Watford after Ashley Fletcher fired in a right-footed shot four minutes from time.
Aston Villa thrashed fourth-tier Barrow 6-0 away, with teenager Cameron Archer netting a hat-trick, while Leeds United left it late at Elland Road before putting three past League One side Crewe Alexandra in the last 11 minutes for a 3-0 win.
Wolverhampton Wanderers beat second-tier Nottingham Forest 4-0 on the road, with all the goals coming after the break.