The Dons striker should have headed home Scott Vernon's cross after 15 minutes but somehow put the ball wide from close range.
Ryan Jack also went close for the hosts with his shot well saved by Marian Kello, but the fierce winds were far from conducive to good football and goalmouth action was subsequently at a premium.
The Dons made two changes with Darren Mackie and Fallon preferred to midfielder Chris Clark and striker Josh Magennis, while winger Peter Pawlett served the second game of a two-match ban.
Hearts, who were missing the injured Jamie Hamill, also shuffled their pack with five changes to the side that started the 2-0 win over Motherwell.
Darren Barr came in to replace Hamill, while there were also starts for John Sutton, David Obua, Andrew Driver and Rudi Skacel.
Ian Black, Stephen Elliott and Mehdi Taouil all dropped to the bench, while Scott Robinson was a notable absentee.
The game started in a scrappy manner with both sides struggling to adapt to the testing conditions.
The visitors carved out the first openings with Sutton seeing his deflected shot drift out for a fruitless corner before David Templeton fired over from the edge of the box.
Youl Mawene picked up the first booking of the match after 13 minutes for ending Marius Zaliukas's rampaging run with a crude challenge.
The Dons should have opened the scoring two minutes later when Fallon somehow headed wide from six yards out with only the goalkeeper to beat.
Hearts almost capitalised when Aberdeen goalkeeper Jason Brown's attempted clearance into the ferocious wind could only reach the edge of his own area, but the Welshman redeemed himself by saving Sutton's left-foot volley.
Aberdeen created a decent opening at the other end just before the break when captain Richard Foster skipped past two defenders before finding Jack, whose shot was well saved by Kello.
Hearts made a change at the break with former Inverness Caledonian Thistle midfielder Black introduced at the expense of Skacel.
The Dons went painfully close from a well worked free-kick after 54 minutes when Kari Arnason drilled a low shot into the side-netting from 25 yards.
The hosts, who brought on Clark for Mackie after 64 minutes, were looking the more likely to take the lead but were too often let down by their final pass.
Arnason's long throws into the box had looked Aberdeen's most promising route to an opener, but the Hearts rearguard coped admirably with the threat.
Both sides, cheered on by the 9,210 fans who had braved the weather, pressed for a winner late on and it was the Dons who almost stole all three points only for Milsom to miss the target with a downward header from Jack's cross in the final minute.