After appearing to be on easy street after going in at the interval 2-0 up, the Dons were made to hang on in the dying minutes when former player Steve Lovell pulled a late goal back.
Aberdeen made three changes from their previous outing with Andrew Considine, Derek Young and Sone Aluko being replaced by Jeffrey de Visscher, Gary McDonald and Chris Maguire.
Jimmy Calderwood's side made their intentions clear as early as the fifth minute. Mark Kerr played the ball wide to Lee Mair whose first-time cross deflected off Darren Barr. The ball looked to be looping over Robert Olejnik who quickly adjusted his feet to brilliantly tip the ball over the crossbar.
The home side went close again on eight minutes when de Visscher fed the ball in to McDonald who turned his marker before driving in a fierce effort which was parried over the bar.
From the resulting corner, Charlie Mulgrew's cross into the area was missed by the Falkirk keeper leaving Alexander Diamond with an excellent chance to open the scoring. However, the centre-back somehow contrived to nod over from four yards.
Minutes later the visitors almost broke the deadlock on the counter-attack. Burton O'Brien played the ball on to Dean Holden whose swerving 30-yard effort was tipped on to the face of the bar and over by Jamie Langfield.
Aberdeen deservedly went ahead on 16 minutes. McDonald played the ball to Miller who held it up for long enough for the midfielder to run on to the return pass.
As McDonald raced clear into the box he was hauled back by Gerard Aafjes. The referee correctly awarded a penalty kick, booking the Falkirk centre-back at the same time. Up stepped top scorer Miller who did his usual immaculate job of coolly stroking the ball home.
In the 30th minute Falkirk strikers Lovell and Michael Higdon combined to set up a great chance for Scott Arfield who, with a clear sight of goal, carelessly smashed the ball high over the bar.
The visitors continued to impress and might have been on level terms in the 35th minutes. Higdon's clever pass released Graham Barrett behind Mulgrew, only for Dons keeper Jamie Langfield to race from his goal and nick the ball away as Barrett was about the slot a shot home.
Aberdeen extended their lead on 43 minutes. From wide on the left Maguire played in a telling ball which drifted away from the outstretched Darren Barr, to find Miller lurking at the far post where he neatly clipped the ball over the visiting goalkeeper and into the unguarded net.
Aberdeen were surging all over Falkirk at this stage as the game approached half-time. Miller was unlucky not to complete his hat-trick when Olejnik brilliantly tipped his 20-yard effort over the bar.
From the corner kick Diamond was just off target with a flashing header which fell the wrong side of the post.
In the 62nd minute ex-Dons favourite Lovell easily skipped past Scott Severin before firing in a low 17-yard effort which was blocked by Langfield.
Aberdeen were inches away from putting the game beyond Falkirk three minutes later.
McDonald drifted away from his marker as he raced on to Mulgrew's pass. He placed the ball across the goalmouth where Mackie somehow failed to get a vital touch.
The visitors countered and threatened minutes later. A clever knock-on by Higdon found Lovell who played the ball across the area to Barrett but the pass was just too heavy and the opportunity was gone.
In the dying minutes of the game poor defending by the Aberdeen centre-backs failed to cut out Holden's cross from the left with the ball falling to Lovell who buried his 14-yard effort into the corner of the net for his seventh goal of the season.