Jason Roberts and Tim Sherwood led the way with two goals each as Third Division Northampton were cut down by Portsmouth.
The Cobblers were up against it once skipper Paul Reid was sent off early on for hauling down Roberts as the on-loan West Brom striker raced through on goal.
Redknapp left out eight of the starting line-up that suffered Pompey's first Premiership defeat of the season against Blackburn on Saturday.
But Teddy Sheringham and Roberts pulled the overworked Northampton defence to pieces.
Northampton were without £150,000 leading marksman Josh Lowe but in their rare attacks looked lively.
Portsmouth were two goals up after 16 minutes after Sherwood headed in Gary O'Neil's corner and then Roberts ran on to a Sebastian Schemmel pass to clip the ball over Lee Harper.
Matthew Taylor struck a fierce 20-yard shot for 3-0 five minutes before the break and Roberts nodded his second and Pompey's fourth just before the hour.
Northampton's first consolation was a twice-taken penalty by Chris Hargreaves in the 76th minute.
Reserve goalkeeper Harald Wapenaar saved the first attempt after bringing down Lawrie Dudfield but was ruled to have moved too early.
Sherwood tucked away his second with a neat header and Dudfield's stoppage-time effort for the visitors was merely academic.
Portsmouth: Wapenaar, Schemmel, Primus, Foxe, Taylor, O'Neil, Sherwood, Smertin, Pericard, Sheringham, Roberts. Subs: Srnicek, De Zeeuw, Stefanovic, Stone, Ayegbini.
Northampton: Harper, Chambers, Willmott, Reid, Carruthers, Youngs, Trollope, Harsley, Hargreaves, Richards, Dudfield. Subs: Thompson, Burgess, Sampson, Asamoah, Rickers.
Referee: D Crick (Surrey).