The Dons must now look for a Hearts win over Dundee United at Tynecastle coupled with a positive result at Ibrox. However, this has not been a happy hunting ground as they have not won in Govan since 1992.
With only four minutes gone Bruno Aguiar's clever ball inside Charlie Mulgrew set up Adrian Mrowiec, whose first-time effort flashed just wide of the upright from 16 yards.
Aberdeen passed up an excellent chance on 15 minutes when skipper Scott Severin failed to hit the target from eight yards after a Mulgrew corner was missed by everyone at the near post.
On 22 minutes the home side threatened again when Lee Miller climbed above Marius Zaliukas to knock the ball into the path of Chris Maguire, the young striker advanced to the edge of the area where he played the ball on to Jamie Smith, the veteran midfielder's left-foot shot on the turn was brilliantly held by Marian Kello.
With 30 minutes gone Mulgrew and Foster combined on the left for the latter to play in a great cross which was headed just over by Miller when under pressure from Eggert Jonsson.
Aberdeen were almost caught on the counter attack on 38 minutes, Calum Elliot latched on to a loose ball to smash in a low shot from 22 yards which went just wide of the upright.
As half-time approached Hearts ought to have broken the deadlock. Substitute David Templeton's quick feet saw him skip past several Dons defenders only for Jamie Langfield to get a vital touch to turn the ball around the post.
The visitors started the second half very much in the ascendancy, with 49 minutes gone a cross from Aguiar was just missed at the near post by Elliot with Langfield getting a touch to flick the ball away from Templeton as the youngster lined up to knock the ball home.
Aberdeen raced to the other end and missed a glorious chance on 54 minutes, Javan Vidal found Chris Maguire racing clear of Zaliukas, his fierce cross-cum-shot was only parried by Kello into the path of Smith, who somehow could only hit the post from seven yards.
Minutes later Sone Aluko burst past two defenders before blasting in a great effort which brought out an excellent stop from Kello.
A bizarre incident on 63 minutes led to Zaliukas receiving a straight red card for an off-the-ball incident which was brought to the referee's attention by the far-side assistant referee John Gilmour.
Aberdeen were becoming increasingly desperate as they sought to blow the race for Europe wide open.
On 76 minutes they wasted another great opening, Richard Foster's persistence saw him break through the tackle of Michael Stewart before teeing up Smith who got plenty of power into his left-foot effort, but agonisingly for the home fans the shot flew the wrong side of the target.