Jerome’s long-range curler 10 minutes from time capped a brilliant Canaries fight-back after Gary Hooper had cancelled out Matt Ritchie’s early opener.
New boss Neil watched most of the match from the stands but came down to join caretaker Mike Phelan on the touchline after Jonny Howson had been sent off midway through the second half.
Phelan picked the Norwich line-up for the final time following Neil’s arrival from Hamilton and his side made a bright start to the game.
Bradley Johnson could have opened the scoring after only four minutes when he headed over from Hooper’s flick.
Midfielder Alex Tettey also had a long-range shot deflected wide as Bournemouth were put under pressure in the early exchanges.
The Cherries are not the nation’s leading scorers for nothing, though, and showed their ruthless streak by putting away their first meaningful chance in the 18th minute.
Marc Pugh teased Steven Whittaker down the left wing and his cross was emphatically thumped home by Ritchie from the corner of the penalty area.
Ritchie had earlier signed a new three-and-a-half year contract keeping him on the south coast until 2018.
Pugh almost doubled Bournemouth’s lead in spectacular style, flicking the ball up on the edge of the 18-yard box before letting fly with a stinging shot, which shaved the crossbar.
Norwich looked threatening on the counter attack and it needed a smart save from home goalkeeper Artur Boruc to block Jerome’s near post shot with his legs.
Norwich levelled in controversial fashion nine minutes before the break with a goal which left the Bournemouth players incensed.
Jerome’s header from a Nathan Redmond corner hit Michael Turner and deflected kindly for Hooper to rifle into the roof of the net.
Bournemouth surrounded referee Chris Foy claiming Turner had handled the ball, although TV replays proved inconclusive.
The second half was a scrappy affair as Bournemouth struggled to get their usual passing rhythm going and Norwich dug in doggedly.
The visitors were forced to play out the final 27 minutes a man light after Howson was shown a straight red card for a two-footed lunge on Bournemouth substitute Yann Kermorgant.
The dismissal prompted new Carrow Road chief Neil to scurry down to pitch-side and his instructions clearly had the desired effect.
Ritchie twice fired wide before Jerome put Norwich in front, the former Stoke man curling an unstoppable right foot shot past a sprawling Boruc from 20 yards.
The defeat ends Bournemouth’s 14-match unbeaten run but Eddie Howe’s side stay top thanks to Derby’s 1-0 win at Ipswich earlier in the day.