The Dons had lost their last five games but Miller lifted the gloom with a power header which lifted the game out of mediocrity.
The match ended controversially with Falkirk defender Lee Bullen sent off by referee Willie Collum for denying Scott Severin a scoring chance.
It looked a harsh decision and Dean Holden and Jackie McNamara were both booked for their protests.
Falkirk looked lively early on and a great cross from the left from Neil McCann picked out Steve Lovell, but his header lacked power and direction and the ball drifted wide.
The swirling wind made conditions difficult for both teams and there was a let-off for Falkirk when Holden was short with a pass back and Scott Flinders made an untidy challenge on Darren Mackie, but referee Mr Collum saw nothing untoward.
Lovell had the ball in the net for Falkirk shortly afterwards but he fouled Severin as he went for the ball and the effort was chalked off.
Scott Arfield then made a great turn in the box from a McCann pass but Severin was alert to the situation and hacked the ball clear.
Severin then gave the ball away to McCann but the winger's cross was too deep and Jamie Langfield collected easily.
At the other end, Bullen gave away possession cheaply in similar fashion which gave Stuart Duff a chance to create something.
He found Miller at the edge of the area but the striker's half-hit shot ran tamely wide with Flinders again untroubled.
Andrew Considine then robbed McCann before unleashing an ambitious 30-yard shot which was always going too high, but at least was the direct effort the Dons had been lacking.
Lee Mair had the first-half's only attempt on target with a header in the final minute which Flinders clutched confidently.
Aberdeen made the breakthrough after 48 minutes. Considine caught the Falkirk defence flat-footed with a great cross from the left and Miller powered a header low past Flinders.
Falkirk almost equalised moments later but Higdon dragged his shot narrowly wide after breaking down the right.
Lovell then had a couple of decent chances in front of goal. He steered a shot wide after a cutback from McCann and then had an even better opportunity, but blazed his shot over the top.
The game erupted in injury time with Bullen's sending-off which saw McNamara and Holden both booked for dissent.