The win sees Town move back to within a point of Premier Division leaders St Neots.
Full-back Tommy Kinch, who made a couple of early-season appearances for the Town on loan from Brackley Town, returned to Communications Park on Tuesday to score
the second goal against Cogenhoe direct from a corner just before half-time.
Meanwhile the Town's Chris Godwin has moved to Bromsgrove Rovers in order to play closer to his West Midlands home.
Josh O'Grady, Tom James and Josh Blake were all relegated to the bench after Saturday's goal-less draw at Stotfold as a much-changed Town team went ahead in the 13th minute when Cooks' goalkeeper George McClellan dropped a Paul Lamb corner to present Berwick with a close-range tap-in.
An Andy Gregory "goal" was ruled out for offside soon afterwards and Berwick was booked for a foul tackle before Kinch doubled the Town's advantage direct from a corner in the 42nd minute.
Substitute James also picked up a yellow card in the second-half and the game was back in the balance at 2-1 in the 75th minute when Alfie Taylor beat the offside trap to follow-up his two goals for Northampton Spencer on the same ground last season.
The pressure was eased again though with Berwick scoring twice inside 60 seconds to take his seasonal tally to 42 in all competitions as the Town followed up a 4-3 win at Cogenhoe in October with another four-goal haul this week.
A trip by Dan Thomason on James enabled Berwick to score from the penalty spot in the 85th minute before rounding the goalkeeper to complete his fifth hat-trick of the campaign less than a minute later.
The 4-1 scoreline flattered the Town but came as welcome relief after the two points dropped at Stotfold last Saturday.
Meanwhile Long Buckby remain on the fringes of the title race following a lone-goal success against Raunds at Station Road.
Buckby's pursuit of second place saw them notch up another win on Tuesday night but bottom-placed club Raunds battled hard throughout the game.
A second minute Jon Crockett goal gave the Bucks all three points.
Daventry United will take on St Ives in the semi-finals of the UCL KO Cup after the Saints' 4-2 beating of Blackstones.
The winners of that game face Daventry Town's conquerors Newport Pagnell in the final.
