It was Pitman who broke the deadlock in the first half at Kingsmeadow after first winning and then converting a penalty.
Chaplin came off the bench immediately following the interval and took just five minutes to wrap the points up.
Kenny Jackett made three changes from the side narrowly beaten at home by Rotherham the previous week.
Damien McCrory was handed a debut at left-back, while there were first starts for both Matty Kennedy and Nathan Thompson.
Chaplin dropped to the bench, while there was no place in the matchday squad for Brandon Haunstrup or the injured Jack Whatmough.
There was also a change of system, with the wing-backs dropped in favour of a more familiar four-man defensive line.
AFC Wimbledon included a couple of familiar names in their line-up, giving starts to former Fratton stars Paul Robinson and Andy Barcham.
A large grey cloud made its way ominously across the sky as the first half progressed, although the action on the pitch was fairly sedate to start with.
There were a couple of half-chances, but certainly nothing of note as both keepers enjoyed a watching brief.
Barcham did enjoy a surging run through the Blues back-line, only to end the move with a wild effort that flew over the bar.
Matty Kennedy then did well to pick out Pitman in the box, but the visiting skipper had strayed narrowly offside.
It was midway through the first period when the first noteworthy opportunity finally arrived, as Dean Parrett curled a shot narrowly past the post.
Adam May was slightly more off target at the other end of the pitch, with his effort on the bounce clearing the stadium.
Pompey threatened more on 31 minutes following some great work by Jamal Lowe to beat his man down the right.
But Pitman slipped at the worst possible moment before recovering and seeing a shot deflected just off target.
The resulting corner was swung in by McCrory and led to an almighty scramble, with Matt Clarke eventually poking wide.
It was the visitors who broke the deadlock on 38 minutes, however, after Pitman was pushed to the ground by Barry Fuller.
Referee Kevin Johnson immediately pointed to the spot and PITMAN dusted himself down to confidently send George Long the wrong way.
A stretching Oliver Hawkins was then only able to send the ball into the keeper’s arms following some good build-up from Kennedy and Lowe.
The home fans were left angry by a couple of challenges just before the break, feeling that Pitman should have been punished for a clash with Robinson that left the defender with a large gash on its head.
There was then a nasty collision between Hawkins and Callum Kennedy that resulted in both players sprawled on the turf.
Johnson decided to blow for half-time and Pompey’s striker was left bandaged and covered in blood, while the Wimbledon full-back had to be stretchered off.
AFC Wimbledon 0
A large flash of lightning and clap of thunder meant that the start of the second half had to be delayed for a few minutes.
Hawkins, unsurprisingly, was not present when the sides did emerge, with Chaplin on to replace him up front.
And it was CHAPLIN who doubled the advantage just five minutes after the restart, showing typical predatory instincts to capitalise on Anthony Hartigan’s fluffed clearance and slot home.
Another substitute almost reduced the Dons’ deficit at the other end, though, with Luke McGee doing just enough to put off Lyle Taylor as he bore down on goal.
Wimbledon were trying to find a route back into the game and Christian Burgess headed George Francomb’s cross clear, while McCrory’s block kept out Taylor’s follow-up.
But the Blues almost extended their lead on 68 minutes, with a powerful free-kick from Pitman superbly kept out by Long.
The visitors were comfortably holding onto their advantage, although Harry Forrester bundled his way into space before McCrory cleared the danger.
Pompey had understandably tempered their ambitions, although an audacious 35-yard lob from May had to be tipped over by a back-pedalling Long.
The hosts’ day soon got even worse when Burgess attempted an over-head kick and inadvertently caught Robinson.
It was clearly accidental and Burgess was booked, but the defender had to go off and the Dons had already made all three of their substitutions.
But, in truth, a comeback always looked unlikely for Wimbledon, as Pompey comfortably saw out the closing stages to secure all three points.
AFC Wimbledon (4-3-3): Long; Fuller (c), Robinson, Nightingale, Kennedy (Hartigan 46); Parrett (Forrester 57), Francomb, Trotter; Appiah, McDonald, Barcham (Taylor 57)
Subs not used: McDonnell, Sibbick, Kalambayi, Kaja
Pompey (4-2-3-1): McGee; Thompson, Burgess, Clarke, McCrory; May, O’Keefe; Lowe, Pitman (c), Kennedy (Bennett 67); Hawkins (Chaplin 46)
Goals: Pitman 38 (pen), Chaplin 50
Booked: O’Keefe, Kennedy, Burgess
Subs not used: Bass, Talbot, Close, Donohue, Naismith
Referee: Kevin Johnson
Attendance: 4,585 (786 Pompey fans)