Pompey were not quite able to cause an Emirates FA Cup upset against Premier League opposition at Fratton Park.
But the hosts can be proud of their efforts against Bournemouth – as well as their attractive passing football.
A shock looked on the cards when Gary Roberts deservedly fired the Blues ahead just before the interval.
But Joshua King and Marc Pugh were on target in the second half to ensure the Blues’ cup adventure ended at the fourth round.
Paul Cook made three changes from the side beaten by Oxford in the league the previous week.
Christian Burgess, Ben Close and Adam McGurk all came into the starting 11, with Adam Webster, Adam Barton and Gareth Evans dropping to the bench.
It was the League Two side who looked more threatening in the opening stages, with Marc McNulty causing plenty of problems.
One of his surges forward was only halted by the presence of former Fratton favourite Sylvain Distin, while a shot moments later flew narrowly wide.
Pompey also won a couple of corners in quick succession that caused problems in the Bournemouth back-line.
The visitors had to rely on Adam Federici to keep the scores level on 19 minutes, with the keeper getting down sharply to tip behind McNulty’s long-range effort.
Bournemouth, unsurprisingly, improved as the game progressed and Eunan O’Kane curled a free-kick wide after Burgess had been booked for a foul on Glenn Murray.
Ryan Fulton then made a super save, getting down low to his left to turn Juan Iturbe’s drive onto the post.
But it was the Blues who finished the half with a flourish and it was a patient passing move that led to them taking the lead on 43 minutes.
The Cherries could not win back possession as their opponents selfishly kept hold of the ball and the move ended with McNulty squaring for ROBERTS to slide home.
Fratton Park erupted with jubilation as the midfielder wheeled away to celebrate and there were more cheers when referee Mike Dean blew for the interval soon after.
Pompey remained on the attack once the second half got started, although Dean was not convinced by Enda Stevens’ optimistic penalty appeals.
Ben Davies then curled a 30-yard free-kick over the wall, but Federici was able to keep hold of the ball.
And Kyle Bennett did brilliantly to skip past Shaun MacDonald’s challenge and feed Roberts, who was almost falling to the ground as he dragged a shot wide.
Bournemouth responded with a double change, with Matt Ritchie getting warm applause from all four sides of the ground as he came on alongside Pugh.
And Cook made two switches of his own, as Roberts and Burgess made way for Evans and Webster.
The latter was immediately involved in the action, making a vital interception to stop King’s low ball reaching its intended target.
But the Cherries were level on 71 minutes when Davies conceded a free-kick wide on the left and O’Kane’s delivery was glanced home by King.
Pompey had a chance to restore their lead through a set-piece of their own moments later when McNulty was felled.
Bennett stepped up to take the free-kick and his shot was claimed right underneath the crossbar by Federici.
Cook made his final throw of the dice on 77 minutes, with Ben Tollitt replacing McGurk on the wing.
Both sides seemed keen to avoid a replay and were not content to simply sit back and defend.
It took a fine challenge from Matt Clarke to stop Lewis Grabban from firing the Cherries ahead from just a couple of yards out on 83 minutes.
But they did score just seconds later when, after Fulton had saved superbly from Grabban, substitute Pugh was on hand to convert.
Pompey tried their best to find a late equaliser, but it was the Premier League side who held on to book their place in Sunday’s draw.
Pompey (4-2-3-1): Fulton; Davies, Burgess (Webster 62), Clarke, Stevens; Close, Doyle (c); McGurk (Tollitt 77), Roberts (Evans 62), Bennett; McNulty
Goals: Roberts 43
Subs not used: Jones, Barton, Naismith, Chaplin
Bournemouth (4-4-1-1): Federici; Smith, Elphick (c) (Francis 77), Distin, Cargill; Iturbe (Pugh 62), O’Kane, MacDonald, King; Grabban; Murray (Ritchie 62)
Goals: King 71, Pugh 83
Subs not used: Allsop, Simpson, Wakefield, Butcher
Referee: Mike Dean
Attendance: 18,901 (2,885 away fans)