The former Australia national team boss Postecoglou saw his side give up a lead again as they slumped to a 2-1 defeat after extra-time in Denmark on Wednesday to lose 3-2 on aggregate and miss out on the group stages for a fourth consecutive year.
Callum McGregor had given Celtic the lead with a stunning strike in the 48th minute but Mabil levelled on the hour mark just after James Forrest had missed a sitter.
Celtic had the better of the majority of the 90 minutes but ran out of steam and offered little response to Raphael Nwadike's strike early in extra-time.
While Midtjylland march on to face PSV Eindhoven, Celtic will travel to the Czech Republic to take on Jablonec in the Europa League third qualifying round next Thursday.
Postecoglou, who was appointed on June 10, three-and-a-half months after Neil Lennon left, expressed frustration over the club's "hesitation" in the transfer market ahead of the first leg and admitted his side would never be as badly prepared for a big match again.
Although Celtic have made five summer signings, two of them are still in quarantine and only 19-year-old Israeli winger Liel Abada featured in Denmark with former Sheffield Wednesday pair Liam Shaw and Osaze Urhoghide left on the bench.
The dearth of options at the back highlighted the challenges facing Postecoglou.
With Kristoffer Ajer recently departed, replacement Carl Starfelt in quarantine, Christopher Jullien still working back from a serious knee injury and Nir Bitton suspended following his needless first-leg red card, Celtic had to rely on youth.
Stephen Welsh partnered 18-year-old Dane Murray in central defence while Anthony Ralston continued at right-back and impressed with his forward play and two crucial second-half blocks.
Greg Taylor, at 23, was by far the most experienced of the back four but the left-back lost his marker for the hosts' equaliser.
Postecoglou dropped Vasilis Barkas after the Greek goalkeeper gifted Midtjylland their Parkhead equaliser and handed Scott Bain the gloves.
Celtic, who generally controlled possession in the first half, went ahead when skipper McGregor chested the ball down and volleyed into the top corner from 25 yards.
But soon after Socceroo Mabil was able to guide a free header past Bain from 10 yards.
The Danes could have won it in stoppage-time when 18-year-old forward Victor Lind got in between the Celtic centre-backs but volleyed wide in space from 12 yards.
Yet they were ahead four minutes into extra-time when Nwadike had time to control Mads Hansen's square ball before finishing high into the net from six yards.