LYNN held the Brewers for the second time this season, but failed to end Nigel Clough's side's unbeaten record.
Darren Stride's superb strike six minutes before the break looked to have given the Brewers an interval advantage.
However, in first half stoppage time, Burton goalkeeper Matt Duke fumbled a Simon Dakin cross over his own line.
Neither side really threatened after the break although Chuck Martini was relieved to see his clearance, which had rebounded off Clough, fly over the bar.
Lyndon Rowland forced Duke into a smart save while Emeka Nwadike blazed a good chance over and Anton Thomas flashed a shot just wide.
Thomas did have the ball in the net after a shot was deflected home on 71 minutes, but he was denied a goal against his former club after being penalised for hand ball.
"It was a good game," analysed Mills. "They probably edged it in the first half and we edged it in the second half.
"We could have pinched it with a little bit better finishing because we had the better chances in the second half.
"They are a good side and it goes to show that we can be a force. We just need to get a few results behind us."
Lynn: C. Martini, S. Dakin, C. Clark, D. Robinson, G. Fuff, J. March, E. Nwadike, G. Mills, A. Thomas, L. Rowland, T. Wright. Subs not used: J. Palmer, A. Bullimore, L. Gibson.
Burton: M. Duke, A. Davies, T. Henshaw, D. Rennie, M. Blount, D. Wassail, D. Stride, J. Kavanagh (P. Lyons 77mins), D. Anderson (D. Holmes 90), N. Clough, N. Glasser. Sub not used: A. Garner.