The Englishman notched the decisive goal after team-mate Christian Nade, in first-half injury-time, cancelled out Aberdeen striker Darren Mackie's superb opener.
Driver's goal arrived following a period of sustained pressure although the woodwork saved Hearts at the death after Chris Maguire's shot cannoned off the upright.
In a high-octane start to proceedings, Aberdeen settled the better of the two teams and had the hosts pegged back.
The home support were relieved to see the sting taken out of Scott Severin's 30-yard free-kick by the wall after Eggert Jonsson went in hard on Lee Miller.
After absorbing Aberdeen pressure, Hearts created their first chance in the 12th minute.
Zander Diamond failed to deal with Robbie Neilson's long ball forward but Jamie Langfield came to his rescue as Nade tried to lift the ball over the goalkeeper.
Aberdeen made the breakthrough in the 19th minute with a superb volley from Mackie.
Charlie Mulgrew delivered a diagonal ball between Jonsson and Janos Balogh from the halfway line and Mackie placed his effort into the net.
Hearts striker Gary Glen made Langfield work with a tame deflected effort minutes later from a Lee Wallace cross.
Hearts were more direct after going behind and, after a driving run from midfield in the 26th minute, Bruno Aguiar unleashed a shot that flew over the bar.
At the other end, Hearts failed to clear their lines and Lee Miller hit a shot direct at Balogh, who needed assistance from his team-mates after only managing to block the effort from the corner of the box.
The naturally left-footed David Obua was looking to cut inside Mulgrew when he had possession on the right flank and the Ugandan did well to create room for a shot from outside the area that was blocked.
In a near-identical move, Obua again cut inside in the 36th minute and his left-footed shot was well held by Langfield.
From an Aguiar corner, Michael Stewart launched a shot goalward that cleared the bar.
Hearts levelled in first-half injury-time through Nade following patient build-up play.
Aguiar's deft cross to the back post was headed back into the area by Obua and the Frenchman angled a header into the corner of the net past Langfield.
The first clear-cut chance of the second period fell to Aberdeen in the 53rd minute when Diamond rose high to meet Mulgrew's free-kick although Balogh got down well to save the header.
Hearts winger Obua then delivered a teasing low cross from the left side of the box but there were no takers.
Hearts manager Csaba Laszlo had to be restrained by the fourth official after Richard Foster, who was subsequently booked, lunged into a challenge on Nade right in front of the dugouts.
The hosts were dominant after the restart and won several corners that they failed to put to good use.
However, the home side eventually made their pressure count in the 66th minute when Obua's cross from the left was headed in at the back post by Driver to send the Hearts support into raptures.
With time running out, Hearts were content to sit on their lead and that encouraged the Dons to attack.
Substitute Tommy Wright was thrown on and he spurned a great chance to level, volleying Diamond's knock down well over from close range in the 80th minute.
Mackie then saw a shot deflected inches wide of the post after getting past Neilson in the box.
Aberdeen were denied an injury-time equaliser by the post after sub Maguire's volley came back off the upright from Foster's cross.