Didier Deschamps' side - who were knocked out of this year's World Cup by eventual winners Germany in the quarter-finals - looked headed for just their second defeat of 2014 with 20 minutes remaining.
Albania were the better side in the first half and claimed a deserved lead when Megrim Mavraj headed home in the 40th minute.
But Albania, who beat Portugal in September and are evidently on the up, were denied another famous win by the Atletico Madrid star.
Griezmann played a neat one-two with the influential Christophe Jallet and cut inside Lorik Cana before curling into the bottom corner from just inside the penalty area in the 73rd minute.
Albania can take plenty of heart from their display, though, as they moved on from the ugly scenes that marred their abandoned clash with Serbia in Belgrade last time out.
Players, coaches and fans clashed as a drone carrying a flag bearing Albanian national symbols was lowered into the stadium during the Euro 2016 qualifier.
UEFA subsequently issued a statement declaring Serbia had been handed a 3-0 walkover win, but had also been deducted three points, while both nations were fined €100,000 - a ruling that they launched an appeal against on Friday.
Football was the talking point on this occasion, though, as Albania - compact at the back and exciting on the counter-attack - caused several problems for the hosts.
Alexandre Lacazette's recall headlined France's squad and he was handed a first international start by Deschamps.
The Lyon man is in outstanding form, having netted 11 times in 13 Ligue 1 appearances this season, but he had little involvement in the early stages as Albania started brightly.
Experienced midfielder Cana - who played over 200 matches for Paris Saint-Germain and Marseille - was unlucky to see his third-minute header come crashing back off the crossbar following a corner.
Raphael Varane nodded over for France at the other end but Albania kept pressing, as Sokol Cikalleshi narrowly missed the target from just inside the penalty area after a free-kick was played short.
Ansi Agolli denied Mathieu Valbuena a tap-in just after the half-hour mark with a superb block and Karim Benzema then headed into the ground - and over - from Jallet's cross.
It was no surprise when Albania took the lead, though, as another well-worked set-piece proved France's undoing five minutes before the break.
Elseid Hysaj controlled a short corner and turned Jallet before crossing for Mavraj, who nodded powerfully past France goalkeeper Hugo Lloris.
France were improved after the break, but were initially thwarted by some stubborn defending, while Ermir Lenjani and Hysaj forced good saves out of Lloris on the counter.
Griezmann's piece of brilliance levelled things up with 17 minutes left, and then Benzema really should have put France in front.
The Real Madrid man fired over from 10 yards after Albania failed to clear their lines, while a fierce Paul Pogba effort went wide.
France's late pressure was not enough for them to grab a winner, though, as both sides shared the spoils - a result neither side could have complaints with.
