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The Daggers, who beat Macclesfield on Saturday, twice led through Michael Spillane and Scott Doe but the Bees refused to roll over and Izale McLeod and Ben May twice brought them level.
May's looping 30-yard strike in the 79th minute caught goalkeeper Chris Lewington unawares to rescue a point for the home side.
Still said: "It's disappointing to concede a goal like that because this is a game we should have won.
"It was a bizarre goal but sometimes these things happen and Chris made some good saves earlier on, it's just one of those things.
"I thought we played really well here and should have won the game but credit to Barnet, they responded well. We battled really hard and took the lead twice with a great display so I'm pleased with a satisfying performance."
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