Jay Rodriguez made the breakthrough in the first half for the Clarets on a day when chances were few and far between.
"That was a real battle," said Howe. "To be honest, I don't think either side got going and that might be a bit down to the weather.
"I was most pleased how we battled out there though - that was the most impressive part of our display."
The win has lifted Burnley into 14th place in the Championship table, while Millwall remain in the drop zone.