Left wing-back Berkoe, on loan from Salford City until the end of the season, fired home after Charley Kendall's cross hit the post and Jayden Luker's follow-up was saved by Bees goalkeeper Josh Keeley.
It was no more than the injury-hit Cards deserved after a tenacious display extended their unbeaten home run to six games, moving them four points clear of the relegation zone.
"We've got 11 senior players injured and it was backs to the wall but right from the start we got after them and tried to dominate like we have recently at home," said assistant manager Ben Turner. "I think we were thoroughly worth the result.
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