Blackpool went into the game as one of only two unbeaten teams in the division - Queens Park Rangers being the other - and seeking their third win in a row.
And striker Fuller gave them a great start as he pounced on a long throw-in to open the scoring.
The lead was short-lived, though, as Grabban fired a left-footed volley past Matt Gilks, before going close with a deflected effort.
Tom Ince, making his first start of the season, should have restored Blackpool's advantage before the break but struck the post after being put through by the impressive Fuller.
But the visitors did retake the lead three minutes after the break as Bishop headed home Ince's whipped delivery.
Robinson, on loan from Liverpool, was sent off after being shown two yellow cards for fouls on tricky winger Ryan Fraser, but Bournemouth struggled to convert their numerical advantage into clear-cut chances.
Indeed the Seasiders pushed for a third late on and when Nathan Delfouneso missed a glorious chance, Paul Ince threw a water bottle at his own dug-out in anger.
The bottle struck a watching supporter, to whom Ince "immediately apologised" according to his assistant Alex Rae.
Referee Oliver Langford sent the former England midfielder to the stands, but his absence did not prevent his side from holding on for victory.
Live text and stats
Saturday, 14 September 2013
07 Pugh (Thomas 71)
01 Gilks yellow card
03 Robinson two yellow cards, red card
17 Basham (Orr 52)
16 Ferguson yellow card
31 Angel (Delfouneso 62)
19 Fuller (Tyson 75)
Ref: LangfordAtt: 9,446
Attempts on target
Attempts off target
90:00+9:52 The referee blows for the end of the match.
90:00+5:02 The ball is swung over by Eunan O'Kane.
90:00+4:24 The ball is sent over by Eunan O'Kane.
90:00+3:48 Free kick taken by Matt Gilks.
90:00+3:27 The official flags Brett Pitman offside.
90:00+0:58 Shaun MacDonald takes a shot and went wide of the left-hand upright.
90:00+0:04 Nathan Delfouneso makes a clearance.
89:58 The ball is crossed by Ryan Fraser.
89:26 Clearance made by Kirk Broadfoot.
89:22 A cross is delivered by Charlie Daniels.
88:46 Ryan Allsop restarts play with the free kick.
88:38 The official flags Tom Ince offside.
87:22 Shot by Lewis Grabban cleared the target.
87:14 Clearance by Bradley Orr.
87:08 Shaun MacDonald crosses the ball.
86:52 Gary MacKenzie makes a clearance.
86:49 The ball is delivered by Steve Cook.
86:02 The ball is delivered by Steve Cook.
85:28 Clearance by Bradley Orr.
85:23 Lewis Grabban sends in a cross.
83:47 Nathan Tyson produces a cross.
82:12 Nathan Delfouneso has an effort at goal and missed to the right of the net.
80:58 Unfair challenge on Gary MacKenzie by Wes Thomas results in a free kick.
80:39 Corner taken by Charlie Daniels from the right by-line.
80:12 Ryan Fraser has an effort at goal.
78:37 Eunan O'Kane takes a shot.
77:33 Gary MacKenzie makes a clearance.
77:30 A cross is delivered by Charlie Daniels.
77:24 Free kick taken by Eunan O'Kane.
76:56 Gary MacKenzie gives away a free kick for an unfair challenge on Wes Thomas.
76:25 Centre by Tom Ince.
76:07 Neal Bishop takes the free kick.
75:48 Lewis Grabban concedes a free kick for a foul on Isaiah Osbourne.
74:55 Substitution Nathan Tyson is brought on as a substitute for Ricardo Fuller.
74:24 Ryan Fraser takes the chance to get an effort at goal. Matt Gilks blocks the ball.
74:14 The ball is sent over by Lewis Grabban.
73:52 Corner taken by Brett Pitman.
73:44 Bradley Orr makes a clearance.
73:40 Elliott Ward produces a cross.
72:19 Header at goal by Wes Thomas goes right of the post.
72:10 Simon Francis crosses the ball.
71:27 Clearance by Gary MacKenzie.
71:23 The ball is swung over by Simon Francis.
71:01 Free kick taken by Shaun MacDonald.
70:54 Unfair challenge on Shaun MacDonald by Isaiah Osbourne results in a free kick.
70:27 Substitution Wes Thomas on for Marc Pugh.
70:04 Caution for Matt Gilks.
69:51 Simon Francis produces a cross.
68:23 Bradley Orr makes a clearance.
68:20 The ball is sent over by Simon Francis.
68:08 Corner taken by Brett Pitman from the left by-line.
67:51 Clearance by Kirk Broadfoot.
67:47 The ball is sent over by Charlie Daniels.
67:29 Clearance by Gary MacKenzie.
67:26 Simon Francis sends in a cross.
66:29 Elliott Ward takes a shot and missed to the right of the target.
66:12 Corner from the right by-line taken by Charlie Daniels.
65:41 The ball is sent over by Simon Francis.
64:40 Brett Pitman takes the chance to get an effort at goal and missed to the left of the goal.
64:28 Gary MacKenzie makes a clearance.
64:24 The ball is swung over by Ryan Fraser.
62:25 Kirk Broadfoot restarts play with the free kick.
61:52 Substitution Angel Martinez goes off and Nathan Delfouneso comes on.
61:33 Free kick awarded for a foul by Lewis Grabban on Gary MacKenzie.
60:28 Charlie Daniels crosses the ball in from the free kick.
59:24 Booking for Barry Ferguson for dissent.
58:17 Sent off
58:17 The referee shows Jack Robinson a red card.
58:02 Free kick awarded for an unfair challenge on Ryan Fraser by Jack Robinson.
57:24 The ball is sent over by Bradley Orr.
56:35 Isaiah Osbourne takes a shot and missed to the right of the net.
56:25 Neal Bishop crosses the ball.
54:20 Free kick crossed by Brett Pitman.
54:08 Bradley Orr challenges Charlie Daniels unfairly and gives away a free kick.
53:37 Clearance by Jack Robinson.
53:30 Corner taken by Brett Pitman.
53:09 Clearance made by Jack Robinson.
53:02 A cross is delivered by Charlie Daniels.
52:54 Corner from the left by-line taken by Charlie Daniels.
52:27 Lewis Grabban sends in a cross.
51:51 Substitution Bradley Orr joins the action as a substitute, replacing Chris Basham.
50:52 Free kick taken by Elliott Ward.
50:49 The referee blows for offside against Ricardo Fuller.
50:23 Clearance by Gary MacKenzie.
50:17 Eunan O'Kane sends in a cross.
49:58 Steve Cook restarts play with the free kick.
49:48 Unfair challenge on Marc Pugh by Tom Ince results in a free kick.
47:43 GOAL - Neal Bishop:Bournemouth 1 - 2 Blackpool Neal Bishop scores with their right foot. Bournemouth 1-2 Blackpool.
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