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Portsmouth went into the game 58 places below their opponents in the English league pyramid but they quickly showed that they were not overawed by the occasion, with striker Marc McNulty proving a frequent threat in the final third.
On 28 minutes, Juan Iturbe also went close to opening his account for the Cherries with a well-struck effort but Pompey stopper Ryan Fulton - on loan from Liverpool - made a fine save to tip the ball onto the post.
That intervention proved pivotal as Portsmouth began to get back into the tie, and shortly before the break, they went ahead through Gary Roberts, who got on the end of a McNulty pass to score his seventh goal of the season.
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Portsmouth returned for the second half eager to extend their advantage, with Ben Davies and Roberts both getting shots in on goal, but after the hour mark, when Matt Ritchie and Pugh were introduced, Bournemouth looked the better side.
Moments after Lewis Grabban directed a shot high over the crossbar, King was able to score an equaliser for the Premier League outfit, heading home from an O'Kane free kick sent in from the left flank.
Despite the equaliser, Pompey continued to show ambition in the final third, but with seven minutes left, Bournemouth completed the turnaround through Pugh, who converted from close range after Grabban had seen an effort well saved by Fulton.
Portsmouth continued to fight until the full-time whistle, but Bournemouth were able to seal the win relatively comfortably as they booked their place in the last 16 of the competition for the first time since 1989.