Bristol City took the lead in the 38th minute through a lovely cushioned header from Tyreeq Bakinson but their lead was short-lived when goalkeeper, Daniel Bentley made a shocking mistake and dropped the ball into his own net just before half-time.
Woodgate's men just about maintain their place in the top six as Barnsley's win earlier meant they could have dropped out of the play-off positions with anything less than a victory.
• Story of the match
Bournemouth started the game in a very intense style. Right from the off they were comfortable in possession and looked to exploit Bristol City down the flanks.
Just two minutes in, Junior Stanislas flashed a low cross into the box that just evaded Dominic Solanke. Lloyd Kelly then fired over from 15-yards out in the 6th minute after an inventive corner routine with Stanislas again being the creator.
Despite their lacklustre start, the Bristol City defence stood firm, maintained their shape and defended every Bournemouth attack rigorously.
After the Robins had weathered an extremely positive Bournemouth storm, they began to settle into the game and create some chances of their own.
Famara Diedhiou had Bristol City's first shot of the game in the 18th minute but his shot just deflected past the post.
Attacking midfielder Kasey Palmer started to pull the strings for the Robins and he looked like Bristol City's main threat. In the 31st minute, he curled a lovely finessed effort just wide after some lovely build-up play which started from his excellent pass forward.
Seven minutes later, Bristol City's pressure paid off and Bakinson broke the deadlock.
Palmer did well down the right flank and darted a ball across the face of goal which eluded everyone. Jack Hunt picked up the pieces and floated in a perfect cross which found the head of Bakinson and he expertly found the top right corner.
After such a lively start from Bournemouth, they had certainly taken their foot off the gas and the concession of the goal woke them up again.
Arnaut Danjuma tested Bentley from distance in the 45th minute and after the resulting corner was only cleared as far as Stanislas, the 31-year-old fired tamely goalwards and just as the ball looked to be heading over the bar, Bentley made a huge mess of the catch and dropped the ball into his own net right on half-time.
The second half failed to live up to the billing after a quite frantic first 45. Both sides lacked any real quality in the final third as they struggled to create any clear-cut opportunities.
Adam Smith and Danjuma found themselves in good positions on the flanks but their final ball was poor on a handful of occasions and prevented Bournemouth from troubling the Bristol City defence.
In a very flat second half with the game delicately poised, Bournemouth looked the most likely to score with Kelly and Solanke looking the liveliest for Woodgate's men.
Robins defender, Tomas Kalas, made an excellent clearance in the 75th minute when Kelly drove in a powerful low cross that looked goal-bound and Solanke had a penalty shout waved away in the 81st minute.
Just as it looked like the two sides were about to settle for a point, substitute Philip Billing rose highest from a corner two minutes from time; he nodded the ball back across goal and Carter-Vickers was there to touch the ball home.
Bristol City tried to push on in the closing stages but their efforts were not rewarded as Nigel Pearson falls to a defeat in his first home league game as boss.
Next up for the Robins is a home tie against Queens Park Rangers on Saturday as they look to get back to winning ways. Bournemouth travel to Preston North End and will be hoping for more of the same in the North West.
• Stand out player
Kelly was one of Bournemouth's brightest sparks this evening and his determination to get forward and cause Bristol City problems was admirable.
The 22-year-old had two shots on goal, 56 touches and created one big chance. His quality in the final third was consistently good and in the second half, he looked like the man to create an opening.
He attended to his defensive duties when called upon and had an all-round positive display this evening.
• Takeaways from the match
Woodgate's Bournemouth have quality in all areas on the pitch. Tonight was by no means their best performance but when you have quality in your ranks, you can win games in all types of circumstances.
The introduction of Shane Long and Billing in the second half proves how much depth they have in their squad and that they have players that can change a game from the bench.
If they continue performing and more importantly - winning, they will have no problems securing a play-off spot.
Although they lost this evening, Bristol City were well in the game for 88 minutes. In the closing stages when both teams attacked, the scoreline could have gone either way and losing to a strong Bournemouth team is nothing to lose sleep over.
Pearson has got his new men defending with passion and heart. They attack with real intent and look to do the right things all over this pitch.
If they continue to play in this positive fashion, they could certainly be a dark horse in the push for the play-offs.