Bishop and Fuller scored their first goals for 10-man Blackpool to clinch victory.
Bishop headed the winner for Paul Ince's Championship leaders after Bournemouth's Lewis Grabban cancelled out Fuller's opener.
The Tangerines had young full-back Jack Robinson sent off early in the second half for two bookable offences before manager Ince also headed to the stands for the final 10 minutes following an exchange with referee Oliver Langford.
With Eddie Howe's men boasting a 100 per cent home record this term and Blackpool unbeaten, something had to give.
The home side's new record signing Tokelo Rantie did not make the bench which gave Brett Pitman and Grabban another chance to shine in attack and the pair almost combined after 30 seconds to open the scoring.
The first goal went to former Stoke frontman Fuller though, as he headed his first goal for the club after seven minutes following a long throw-in.
But in an end-to-end first 45 minutes the lead did not last long and Grabban continued his fine form to level the scoring after 12 minutes.
The former Millwall striker latched on to an Elliott Ward flick-on to fire a left-footed half-volley past Matt Gilks for his fifth goal of the season.
Grabban, who has scored in all three of Bournemouth's home league games so far, went close to netting again but saw his 20-yard drive deflected over the bar by Gary MacKenzie.
England Under-21 international Tom Ince should have scored Blackpool's second after 35 minutes but the former Liverpool man could only drag his effort against the post after being sent through one-on-one by the impressive Fuller.
The youngster made up for his miss straight after the restart, whipping a corner straight onto the head of former Notts County player Bishop to score his first for the club after 48 minutes.
Ince's men looked in control, but soon after Isaiah Osbourne had shot narrowly wide they lost on-loan Robinson, the player borrowed from Liverpool given his marching orders after picking up his second booking for fouls on tricky winger Ryan Fraser.
With their opponents down to 10 men after 58 minutes, Bournemouth went in search of the equaliser but struggled to find a way through.
In a pulsating encounter, Blackpool were going for a third goal rather than sitting back and Nathan Delfouneso missed a gilt-edged chance to secure the three points when he sliced his shot wide.
The dreadful miss infuriated his manager, with Ince launching a water bottle in anger towards the back of the dugout. As a consequence, he was instructed to watch the remaining seconds from the stands.
Wes Thomas went close for Bournemouth before Pitman thought he had rescued a point with the final kick of the game but saw his stabbed shot ruled out for offside.