Mensing opened the scoring from the spot five minutes into the second half after James McCarthy had been brought down, coolly lifting the ball high into the net.
Offiong took his tally for the season to five with a superb effort from an acute angle 13 minutes from time - but was sent off in stoppage time for a second yellow card.
Aberdeen saw their flu-stricken first-team stars recover in time for their trip to Accies, with manager Jimmy Calderwood having to make only one change to the side that recorded a fourth consecutive win last week.
Lee Mair, himself one of seven players who had been ill as the Pittodrie side cancelled training for two days this week, started in place of Charlie Mulgrew, who has a knee injury.
Hamilton made two changes from the side that lost at St Mirren last week, with Mark McLaughlin returning from injury and David Graham being promoted from the substitutes' ranks to start in place of Chris Swailes and Tony Stevenson, who has moved on loan to Clyde.
Accies had their first sight of goal after four minutes when Martin Canning rose at the left-hand post to head into the arms of goalkeeper Jamie Langfield from a Mensing free kick.
Jeffrey de Visscher then ballooned the ball over the bar from close range, but atoned inside a minute with a powerful drive that Tomas Cerny had to stretch to tip over the top right-hand corner of the woodwork, Mair heading over from the resultant corner.
Hamilton responded with Offiong putting a 12-yard shot from the centre of goal over the bar after McCarthy's thrilling burst up the left flank and subsequent pinpoint cross.
The action then lulled after that pacy start, however, with Scott Severin's free-kick, which rose just too far over the bar, the only other real chance of the opening quarter.
Graham had the best chance of the half nine minutes from the interval - as Aberdeen attempted to clear their lines, the ball was booted only as far as the midfielder on the right edge of the box, and he instantly unleashed a diagonal drive that whistled just past the opposite corner.
Aberdeen forced a series of throws and corners as the interval approached but despite the vocal encouragement of a huge travelling support, were unable to break the deadlock, Mark Kerr rolling a stoppage-time effort just wide.
De Visscher again came close to the opening goal 35 seconds after the restart, only to put his close-range snap shot past the left-hand upright, and shortly afterwards Severin rose for a header that he sent into Cerny's arms.
Accies then took the lead from the penalty spot after Zander Diamond halted McCarthy's run by bringing down the teenager on the edge of the area - and team-mate Mensing stepped up and scored a cool first goal of the season by lifting the spot-kick high past Langfield into the roof of the net.
Lee Miller had a close-range header put out for a corner, from which Cerny comfortably caught Sone Aluko's delivery.
The Aberdeen striker then forced the home goalkeeper into another impressive save from a dubious 20-yard free kick, awarded after Cerny had rushed from his area to boot clear - Miller struck the ball powerfully but Cerny stretched to push it over the bar.
Tempers became increasingly frayed as the game reached the hour mark, with numerous players steaming in angrily in the wake of a poor Offiong tackle, and referee Stephen Finnie struggled to keep control as a series of rash, crunching challenges flew in from both sides.
Aberdeen pressed for an equaliser in the final quarter of the game, and were denied twice from a De Visscher corner on the left - first Mair's header was headed away just short of the line, and when De Visscher played the ball back in, Miller sent it just off target.
Hamilton defended in numbers but could have doubled their advantage 15 minutes from time when Graham counter-attacked with a thrilling run down the right that took him all the way into the area; Langfield parried his shot and it was hastily booted out before Offiong could follow up.
Offiong made it 2-0 a minute later, however, heading down a throw on the left that was flicked on by McCarthy before turning his marker on the left side of the box and firing an unstoppable angled shot into the opposite side of the net.
The English striker could have doubled his tally four minutes from time when he burst from midfield, holding off marker Diamond to try his luck with a drilled shot that sneaked just wide.
Offiong's fortunes changed in injury time when he was sent off after receiving a second yellow card for kicking the ball away.