Alexander joined on loan from Crawley on deadline day and took less than half an hour to open his account.
However, another new signing, Michael Symes of Albion, ensured the points were shared.
The Dons pressed for a winner after the break but despite their opponents going down to 10 men when Zander Diamond was sent off, it ended all square.
Alexander opened the scoring with a simple finish after Pim Balkestein's lofted free-kick found him unmarked in the six-yard-box after 27 minutes.
Symes, also a deadline day signing from Leyton Orient, levelled the scores nine minutes later with a shot from the left side of the area.
Alexander went close again 10 minutes after the break and was then denied a golden chance to make it 2-1 when Toby Ajala ran the length of the pitch but failed to pick out the unmarked new signing.
Burton were then dealt a blow when captain Diamond was sent-off for head-butting Balkestein while forward for a 76th minute corner.