Homegrown striker Rodriguez opened the scoring in the 31st minute and John Utaka levelled the score two minutes later before Marney's superb overhead kick eight minutes before half-time proved decisive.
Burnley made one change to the side which drew 0-0 at Scunthorpe on Saturday, with Steven Thompson replacing Ross Wallace in the starting line-up.
The Clarets were the better side during the opening exchanges and a cross from Wade Elliott in the third minute was calmly collected by Jamie Ashdown.
Midfielder Jonathan Hogg made his first start for Portsmouth after signing on loan from Aston Villa until the end of the season earlier today.
In the eighth minute, Pompey threatened when midfielder Liam Lawrence collected possession and hit an 18-yard shot over the crossbar.
Four minutes later, an inswinging corner from Daniel Fox was greeted with a firm header from Michael Duff which drew another comfortable stop from Ashdown.
Yet moments later Pompey signalled their threat when Kanu collected possession 20 yards from goal and his shot drew a routine save from Burnley goalkeeper Lee Grant.
Home midfielder Hayden Mullins then saw a long-range strike flash just wide of Grant's left-hand post but Burnley survived and opened the scoring in the 31st minute.
Fox was given time and space on the left flank to cross for Rodriguez to fire home a right-footed shot from six yards out.
Yet the response from Portsmouth was admirable as they drew level within two minutes thanks to a piece of magic from Kanu.
The former Arsenal and West Brom frontman showed superb skill to turn a Clarets defender and create the space for Utaka to rifle in a right-footed shot which flew into the top corner from 18 yards.
Four minutes later, Burnley restored their lead when Thompson's right-footed shot was parried and Marney followed up with a superb right-foot overhead kick from 12 yards.
Eight minutes after the restart, Burnley winger Chris Eagles fired a 20-yard shot just wide of the target.
With 23 minutes remaining, Rodriguez went close to claiming his second but his header was tipped round a post by Ashdown.
Burnley defended everything that was thrown at them by the hosts but they also threatened to claim a third goal on several occasions.
With 12 minutes remaining, Thompson met a cross from Elliott but his header flew wide from 12 yards.
Utaka then saw a long-range shot flash just wide as they went in search of an equaliser but Burnley held firm.