Hibs had an early scare when Rob Jones was caught on the ball and Collin Samuel pounced, but his cutback was cut out by the alert Shelton Martis.
Ivan Sproule teased Christian Kalvenes before thumping the ball across goal and Fletcher just could not get enough control on his finish.
But it was United who snatched the lead after 24 minutes. The lively Craig Conway put over a threatening cross from the left and the unfortunate Martis headed the ball over Zibi Malkowski's head.
United almost scored a second moments later when David Goodwillie was allowed a free shot, but Malkowski beat the ball out at his post.
Conway carved out a chance for himself in 31 minutes, but Jones managed to get in a great block just when it looked as if a second goal looked certain.
The equaliser came on the stroke of half-time when Jones headed in from ten yards after Chris Killen had nudged on a Kevin Thomson free-kick.
It was Fletcher who settled it after 73 minutes with a 16-yard shot after the ball had broken to him off substitute Abdessalam Benjelloun.