Considine was in the right place at the right time to prod home a Ryan Hedges corner in the first half, while Cosgrove climbed from the bench to make the points safe with a clinically converted penalty in the dying embers.
The hard-fought three points sent Derek McInnes' men above Motherwell and into third place in the Ladbrokes Premiership.
Livingston will feel a sense of disbelief after failing to claim at least draw. Scott Robinson and Chris Erskine were both clean through on goal in the first half, while Jon Guthrie rattled the goal frame as they chased a leveller after the break.
However, the Reds held firm for a gutsy, if not comprehensive, victory.
A well-publicised fitness crisis had already robbed McInnes of Scott McKenna, Ash Taylor, Funso Ojo, Craig Bryson and Stephen Gleeson for a testing visit to Almondvale.
And things went from bad to worse when Cosgrove was unable to shake off a slight hamstring complaint, restricting the Dons' attacking talisman to a place on the bench. Curtis Main came in for his first start since a 1-0 defeat at St Mirren on August 11.
Livingston, meanwhile, replaced top-scorer Lyndon Dykes with Robinson following the towering Australian's red card against Rangers last week.
It was Robinson who passed up a glorious opportunity to give Livi the lead with 19 minutes on the clock. Steven Lawless found the former Hearts player with an intelligent reverse pass inside the box, only for Joe Lewis to make a superb block at his near post.
Livi were playing some slick attacking football and another wonderful through ball - this time from Robinson - sent Erskine haring down on goal, only for the winger to curl his effort over the bar.
Gary Holt's charges were punished for their wastefulness when the visitors opened the scoring on the half-hour mark. A Hedges corner was flicked on by the otherwise ineffective Main, allowing Considine to poke home from close range.
Despite boasting a narrow lead, the visiting supporters grew increasingly frustrated by the lack of ambition shown by their side after the break - with countless long passes towards Main being met by groans from the Red Army.
Livi almost breached the Aberdeen barricades with 20 minutes remaining. The normally reliable Lewis could only tip a looping cross to the feet of Jon Guthrie, but the Lions defender saw his effort hit the bar then post and somehow bounce to safety.
However, it would be Aberdeen who rippled the net again.
Cosgrove, fit enough to come on as a substitute, was fouled in the box by Keaghan Jacobs as the Dons counter-attacked the gung-ho Lions. The big striker dusted himself down to slot home his 11th goal of the season.