Falkirk were ahead at half-time thanks to Scott Arfield's 26th-minute strike but Accies came roaring back after the break with two goals in eight minutes that looked set to send them nine points clear of their opponents at the foot of the SPL.
Top scorer Mickael Antoine-Curier headed in his seventh of the season for the Lanarkshire men in the 50th minute and, when referee Dougie McDonald adjudged the Frenchman was brought down by Brian McLean, Simon Mensing put Billy Reid's outfit ahead for the first time in the match.
Their lead lasted only three minutes, though, as former Liverpool youngster Flynn volleyed in what could prove to be a vital equaliser on the hour mark to take Steven Pressley's side to within three points of second-bottom St Mirren.
The match did not take long to burst to life as neat Hamilton interplay in the fifth minute set up Antoine-Curier for a shot which was deflected into the arms of Falkirk keeper Robert Olejnik.
Seconds later at the other end, Flynn capitalised on a mistake from Mensing but found Tomas Cerny equal to his shot from a tight angle.
Bairns defender McLean gave away a controversial penalty in the previous weekend's 2-1 defeat to St Johnstone for hand ball and would have been relieved a minute later to see referee McDonald waved aside the home team's appeals when he appeared to block Flavio Paixao's attempted cross with his arm.
In the 10th minute it was the arm of keeper Olejnik which proved crucial as Dougie Imrie curled in a left-foot effort from the right side of the box which the visiting keeper turned round for a corner.
Three minutes later, Accies full-back Brian Easton beat Darren Barr down the left flank but Olejnik spectacularly tipped Antoine-Curier's header away from his top corner.
Having soaked up 25 minutes of Accies' pressure, Falkirk hit their hosts with a sucker punch.
A long clearance from a Hamilton corner sent Flynn away and the winger's reverse pass presented Arfield with a golden chance.
The Scotland Under-21 midfielder saw his first shot blocked by Cerny but made no mistake when the rebound fell kindly back at his left foot.
It was clear how much the goal meant to Bairns boss Pressley, who performed a jubilant jig on the pitch before being spoken to by referee McDonald for delaying Easton as he tried to take a quick throw-in in front of his dugout.
There was a let-off for the home side minutes later as the officials missed Cerny appearing to reach out of his box in preventing a corner.
With two minutes to the break, Falkirk almost grabbed a second following confusion in the Accies defence.
Martin Canning ducked under Olejnik's long ball but failed to realise Pedro Moutinho was lurking and the Portuguese striker would have been disappointed to see his lob drift just wide.
Hamilton could perhaps count themselves unfortunate to be behind at the break but they enjoyed the perfect start to the second period as they snatched the equaliser after just four minutes of the half.
Vitor Lima needlessly gave away a corner and when Paixao swung the set-piece over Antoine-Curier popped up at the back post to head in from close range after Olejnik had been left rooted to his line in a crowded six-yard box.
The goal rejuvenated Accies and spurred them forward in search of another.
They were perhaps fortunate to then escape when Canning appeared to handle Flynn's attempted cross but they got the second they sought in the 58th minute.
Mensing threaded a pass through to Antoine-Curier 12 yards out and referee McDonald adjudged McLean to have brought the Frenchman down as he tumbled theatrically.
Mensing himself despatched from the spot but Accies' lead was to last just two minutes.
Moutinho sent in a free-kick from the left and, when the home defence failed to deal with his delivery, Flynn rifled his volley low past Cerny.
Antoine-Curier stung the palms of Olejnik with a fierce 25-yard volley with 19 minutes remaining as the Lanarkshire men pressed hard for the goal which would ease their relegation worries.
However, it was the Bairns who almost grabbed the lead for the second time in the match.
Colin Healy's superb through ball sent Moutinho scampering clear but the striker took too long to get his shot away and when he did Canning got back to block.
Accies poured forward in search of the winner in the closing minutes but brave blocks from Tam Scobbie and McLean helped Falkirk hang on for a crucial draw.