Boro hit back from 2-1 down to lead 3-2 and Britt Assombalonga had the chance to increase that advantage from the penalty spot – but he missed.
And Collins made them pay five minutes from time when he swept in a loose ball to ensure a 3-3 draw for the Hatters on their return to the second tier of English football.
They were ahead though with just seven minutes on the clock, Fletcher rising highest to nod home Marvin Johnson's inch-perfect cross.
Town were level in the 17th minute when captain Sonny Bradley, who did not score at all last season, hammered a magnificent 22-yard volley beyond goalkeeper Darren Randolph and into the roof of the net.
Hatters goalkeeper Simon Sluga pulled off a top-drawer save to tip Johnson's curling attempt over the bar – but he was beaten moments later in embarrassing fashion.
Shinnie gave the ball away and it was fed to Assombalonga whose tame effort went through the keeper's grasp and into the net for 2-2 with just eight minutes of the half remaining.
Callum McManaman ought to have put Luton back in front, volleying off target at the back post from Collins' cross – and Collins thought he had made it 3-2 in the opening minutes of the second period, only to see his goal ruled out for handball.
McNair's cross-shot clipped the top of the bar, Lewis Wing thundered a shot high and wide of Sluga's post, while the home side's Dan Potts had appeals for a penalty turned down after being clipped inside the box.
Wing blasted Boro ahead after 69 minutes, beating Sluga with a swerving drive after some clever build-up play.
However, Assombalonga fluffed a great chance to put the game to bed with nine minutes to play, skying his spot-kick after Cranie fouled Hayden Coulson in the area.
Sluga saved bravely from the forward and Town went on to make the visitors pay.
Harry Cornick's cross found Potts, and although his shot was blocked, Collins was on hand to level the scores.
The Hatters thought they had won the contest in stoppage time – but Collins was denied superbly by Randolph.