Derek McInnes' side were aiming for a seventh straight win after booking a Betfred Cup final place against Celtic with a 2-0 win over Championship side Morton at Hampden Park on Saturday.
However, D'Acol scored from the spot in the sixth minute after referee Willie Collum adjudged him to have been manhandled by Pittodrie defender Anthony O'Connor.
The Dons huffed and puffed thereafter but to no avail and the home side's first win in six games left them in sixth place.
Player-manager Martin Canning will be glad of the respite while Aberdeen have to get back up for the visit of leaders Celtic at the weekend.
It was all quite tepid at the SuperSeal stadium until Accies took the lead.
There did not look too much in it when D'Acol went to ground inside the Aberdeen box under pressure from O'Connor as they battled for Dougie Imrie's throughball but referee Collum pointed to the spot.
The 30-year-old Brazilian forward was unconcerned about the visitors' protests as he picked himself up and slammed the ball past keeper Joe Lewis from the spot for his fourth goal of the season.
Aberdeen flapped for several minutes and D'Acol hit the side-netting with a powerful drive from 25 yards before attacker Eamonn Brophy nipped in ahead of O'Connor to toe poke the ball inches past the post.
The Granite City club, however, worked their way into the game and in the 27th minute Accies keeper Gary Woods had to make a good close-range save from Jonny Hayes after the Dons winger had linked well with James Maddison.
With five minutes remaining of the first half, Maddison's free-kick from wide on the right reached the back post but defender Mark Reynolds' unconvincing header was saved by Woods on the line.
Aberdeen appealed for a penalty early in the second half when midfielder Darian MacKinnon tangled with Kenny McLean in the box but Collum stood firm as the away fans behind the goal voiced their displeasure.
The Pittodrie men stepped up the pace.
Striker Adam Rooney headed a Hayes cross straight into the arms of Woods before Hayes fired over on the angle as the Dons fans turned their frustration on Collum.
McInnes went for broke with a triple substitution in the 64th minute with Miles Storey, Niall McGinn and Peter Pawlett replacing Reynolds, Burns and Maddison.
However, the Lanarkshire outfit remained resolute and indeed began to fancy their chances of a second goal as the clock edged towards the 90-minute mark.
Lewis saved from Greg Docherty and Grant Gillespie following Accies breaks and despite a late flurry Aberdeen could find no way through.
Source: DSG