Gareth Seddon was the hero for Kettering for the second round in a row as his strike earned a 2-1 win over plucky fellow non-leaguers Eastwood.
Seddon, whose goal disposed of Notts County in the previous round, gave the Poppies a two-goal lead at Rockingham Road which was good enough for victory despite a late Eastwood fight-back.
Badgers defender Paul Robinson gave the visitors hope with half an hour to go after Seddon and Craig Westcarr had put Kettering in control but the equaliser was not forthcoming.
It was Kettering who applied the early pressure on their opponents, two leagues below them, with Seddon forcing Eastwood stopper Shane Redmond into an early save.
And after 19 minutes winger Richard Graham crossed for Westcarr to power his volley into the roof of the net.
Graham, on loan from Dagenham & Redbridge and making his first appearance in Kettering colours, was allowed far too much space to pick out Westcarr and the ex-Nottingham Forest man made no mistake.
Seddon grabbed his 14th goal of the season soon after break, keeping his cool in front of goal to slot home, with Graham again the provider,
But just a minute later the Badgers gave themselves hope when defender Robinson fired past Poppies keeper Lee Harper from close range.
Eastwood substitute Matthew Rhead had the visitors' best chance to equalise but he could not direct his late header goalwards as Kettering booked their place in the fourth round.