The visitors went ahead in the 15th minute when Billy Clarke converted Rob Purdie's left-wing free-kick.
Claude Gnakpa headed home Rossi Jarvis' free-kick for the equaliser in the 31st minute, but substitute Greg Blundell fired a 90th minute winner.
There was still time for the hosts to hit the crossbar through Asa Hall in stoppage time as Darlington hung on.
The hosts started the game with just one up-front, inviting the visitors to dictate possession early on and leading to a Darlington opener in the 15th minute.
Purdie sent a free-kick from the left towards the near post and Clarke made the most of a hesitant Hatters defence to clip the ball over the advancing Conrad Logan from six yards for his eighth goal of the season.
Luton almost drew level in the 20th minute when Hall let fly with a speculative 35-yard effort but the ball flew just past the post.
The equaliser did arrive though in the 31st minute when Gnakpa picked himself up after being fouled on the right touchline and powered home a bullet header into the top corner from 15 yards after Jarvis' free-kick.
Teenage left-back Jake Howell, making his full home league debut for the Hatters, capped an impressive first half by teeing up Chris Martin eight minutes before the break, but the former Norwich man saw his 12-yard effort well blocked by Simon Brown.
Darlington, who had looked threatening going forward without creating clear-cut chances, fashioned a half-opening two minutes later but Logan saved comfortably after Clarke's turn and 20 yard shot.
Luton made a double substitution just before the hour mark as Ryan Charles came on for Ed Asafu-Adjaye and Marc Pugh replaced Garreth O'Connor and the new 4-4-2 shape almost brought a goal when Pugh crossed from the left for Jarvis to volley at Brown from ten yards in the 66th minute.
The visitors still looked dangerous on the break and Purdie left Harry Worley standing three minutes later as he cut inside from the right after a 40-yard run and fizzed a left-foot effort off the top of the crossbar from 25 yards.
Liam Hatch then missed a great chance in the 75th minute, when he nodded wide from six yards after Clarke's cross from the left and Logan had to produce a smart save to palm aside substitute Blundell's angled drive nine minutes from time.
There was nothing Logan could do to stop Blundell hitting the winner in the 90th minute as the forward's snapshot found the roof of the net from 14 yards after Luton had failed to clear a corner.
Town piled forward for another equaliser and Hall hit the crossbar in the second minute of stoppage time with a left-foot drive from 16 yards.