Dale Stephens opened the scoring in the 50th minute when he collected a pass from Chris Taylor and fired home from 18 yards.
Yet Romain Vincelot headed home from 12 yards before Dagenham's Peter Gain saw red for a challenge on Oumare Tounkara.
Still said: "It doesn't matter what people throw at us, we just keep going and this is a fantastic point away from home.
"We kept Oldham quiet in the first half and when they came out strongly after half-time we adjusted and worked ever so hard.
"I'm pleased with a point because they put us under a lot of pressure, although most of their efforts were from long range. Our back four were magnificent in difficult and slippery conditions and our wide players caused problems by running at people."
Source: PA
Source: PA