The winner came on 40 minutes, when Craig Barker got forward to find the net with a well-placed shot. Popley almost found an instant equaliser, Steve Austen hitting the post, but they trailed at the break.
Popley goalkeeper Andy Grice did well to save from Steve Harvey and Nick Wilcock had a goal ruled out for offside, while Popley's Michael Thomas was denied an equaliser by the assistant's flag.
The only other league match to survive the weather saw Overton United B host Sherborne St John Reserves at Berrydown in Division Two.
Aiden Blackwell put struggling Sherborne ahead from the penalty spot after Gregor Restell brought down Rory Hatherell and the lead was doubled before half-time, Alex Izzo putting Hatherell through to make it 2-0.
Darren White pulled a goal back for Overton with 10 minutes to go, while Restell had a chance to atone for his earlier error in conceding the penalty - but he could only blaze over the crossbar.
The win was only Sherborne's second of the season and moved them off the bottom of the table.