The home team had in-form striker Blly Kee sent off before the break but the visitors posed little threat until substitute Eastwood struck with a deflected 15-yard shot that bounced in off a post..
Referee Tony Harrington dished out yellow cards to Kee and Southend's debutant defender Rob Kiernan inside two minutes to set the tone for a tetchy and uninspiring contest.
Burton defender Shane Cansdell-Sherriff wasted a free header by planting it straight at keeper Daniel Bentley, while Adam McGurk was unable to make the most of Kee's flick-on from close range.
Southend's first chance came after 31 minutes when Barry Corr's firm header tested home keeper Jordan Pickford, who produced an even better save to deny Cauley Woodrow as the visitors posed a threat at set pieces.
Kee received his marching orders a minute before the break for lunging in on Ben Coker and picking up a second yellow.
Woodrow had the Blues' first chance to make their extra man count but saw his close-range shot deflected over in the 75th minute.
Alex MacDonald went close with a header and substitute Matt Palmer hit a post as Burton refused to sit back - only for Eastwood to steal the points when a stalemate seemed the likeliest outcome.