A composed penalty in the first half and a spectacular drive from distance in the second helped the Clarets return to winning ways following last week's home defeat to Wigan.
Geovanni, who thought he had equalised for Hull before then with a free-kick which the referee ruled out, was then sent off to cap a miserable week for the visitors.
The opener arrived in the 19th minute when the referee gave a soft penalty and Alexander found the net from 12 yards. The spot kick came when Stephen Hunt clumsily brought down Tyrone Mears after a fine pass from Wade Elliott.
The Hull players surrounded the referee but Alexander kept his cool and blasted the ball into the bottom-left corner of Matt Duke's goal to give the hosts the lead.
Hull had a goal ruled out in the 67th minute for pushing in the wall when Geovanni curled home a free-kick.
And just a minute later the Tigers midfielder was given his marching orders for a clumsy challenge on Steven Fletcher which saw him pick up his second yellow.
The Clarets pegged their struggling opponents back with Chris Eagles orchestrating much of their attacking work down the right.
And a second, which was always on the cards following Geovanni's dismissal, came in the 77th minute when Alexander fired home in spectacular fashion from 25 yards.
Substitute Kevin McDonald found the evergreen midfielder in space, he advanced completely unchallenged and unleashed a powerful drive which nestled in the bottom-corner of Duke's goal.