Not only does this end the Saints' losing streak, it also saw them move off the bottom of the Premiership.
Former Wallaby Drahm not only slotted the winning drop-goal, but a minute earlier had also kicked a crucial penalty – following two earlier penalties in the second half, after he replaced an out of sorts former England international Paul Grayson at the break.
Drahm stunned a 17,364 Madejski Stadium crowd – the Premiership's biggest attendance this season – with a virtuoso display after coming on.
His match-winning drop-goal arrived four minutes into injury time and left Irish demoralised.
The Exiles led for most of the match, indebted to 18 points from fly-half Mark Mapletoft, including three drop-goals, but Drahm ultimately took top billing.
Saints scored the game's solitary try through scrum-half Mark Robinson.
They now face Leeds on New Year's Day, and after climbing above relegation rivals Worcester, another victory would provide another major survival boost for new coach Budge Pountney.
Saints had gone nine league games without a win, a sequence stretching back to mid-September, but they displayed admirable character when it mattered and found an unlikely hero in Drahm.
Drahm kept them in the hunt with two penalties, yet Irish looked to have done enough when Barry Everitt, Mapletoft's replacement, slotted a 77th-minute drop-goal.
But that was the cue for Drahm to seize his moment, and he clinched the points for Saints with a penalty that took his team to within striking distance, then landed that priceless drop-goal.
The struggling Saints received a triple boost when internationals Ben Cohen, Bruce Reihana and Corné Krige were all passed fit, with England's Cohen lining up at outside centre.
Flanker Andrew Blowers though, had still not recovered from concussion sustained during Northampton's Heineken Cup defeat against Toulouse two weeks ago, so former Bristol Shoguns skipper Ross Beattie made his league debut in the back-row.
Irish included Scottish international Roland Reid at No.8, with Scott Staniforth and Mike Catt forging a high-class midfield partnership.
Cohen was immediately pressed into defensive duties, but cleared the danger impressively after Irish flanker Paul Gustard charged down a Grayson kick. Gustard collided with an upright, trying to chase down Cohen, yet resumed following treatment.
Grayson regained his composure to boot Saints 3-0 ahead through an eighth-minute penalty, which Northampton deserved on the early balance of play, and it was their own poor discipline that allowed Irish a chance to draw level.
Mapletoft's deft midfield break split Saints open, but the diminutive playmaker still made Saints pay, rifling over a 30-metre penalty following a reckless late challenge by Northampton wing Wylie Human.
Mapletoft then missed an angled 25-metre chance to put Irish ahead. Saints though, found themselves under increasing pressure, and further indiscipline cost them.
Cohen was penalised by referee Tony Spreadbury for not releasing possession after diving on the ball in defence, and when he committed a similar offence just two minutes later, Spreadbury predictably sin-binned him.
Mapletoft kicked the resulting penalty, yet Irish could not make their temporary one-man advantage count.
With Cohen off, Northampton claimed a sparkling opportunist try when Robinson grasped Mapletoft's pass on halfway and sprinted 50 metres, diving over despite a despairing tackle from his opposite number Paul Hodgson.
Grayson converted, and although Mapletoft completed his penalty hat-trick during first-half injury time, Northampton shaded a tense opening period where it mattered most – on the scoreboard.
Saints made an interval switch, replacing Grayson with Drahm, and Irish regained the lead on 44 minutes as Mapletoft landed a 35-metre drop-goal.
Mapletoft's strike gave Irish momentum, and with Robinson – Saints' most influential player – struggling from a knock that eventually forced him off, cracks started to appear in the Northampton defensive line.
A second Mapletoft drop-goal, after his initial weaving run through heavy defensive traffic, compounded Saints' degree of difficulty as the game entered its final quarter.
And when he completed his hat-trick just six minutes later, Saints appeared down and out until Drahm gave them a fighting chance by kicking two long-range penalties.
Drahm's accuracy – his third penalty arrived two minutes after Everitt's drop-goal – then guaranteed an exciting finish that reached a shuddering crescendo deep into stoppage time.
For London Irish:Pens: Mapletoft 3DGs: Everitt, Mapletoft 3
For Northampton Saints:Try: RobinsonCon: GraysonPen: Drahm 3, GraysonDG: Drahm