The hosts' best chances fell to Mark Arber and Josh Parker but Town goalkeeper Matt Duke saved their headers at the end of either half.
The win lifts Northampton up to 19th, five points above the relegation zone and six clear of the bottom-placed Daggers.
VIEWS FROM THE DRESSING ROOM
Northampton Town manager Aidy Boothroyd tells BBC Radio Northampton:
"It was a very, very difficult game. We had to rely on our goalkeeper in the first half and he made a fantastic save. I'm delighted we've managed to get the 1-0 and defend as well as we have for long spells.
"We had to stand up to the way they play. We got a couple of breaks and thankfully we took one.
"Our supporters were absolutely fantastic. What a brilliant turn-out and they made noise from start to finish.
"We're not going to get excited with 19th. We've got to finish the season as strongly as we can and we're not out of trouble yet."
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