Danny Swanson broke the deadlock for United after 39 minutes with a cool finish to round off a well-worked move.
However the talking point of the match came three minutes earlier when the home side were awarded a penalty for a foul on Prince Buaben by Chris Hogg on the edge of the area. Referee Steve Conroy initially appeared to point for a free-kick before deciding it was a penalty, but Graham Smith came to Hibernian's rescue by palming away David Goodwillie's weak spot-kick.
Hibernian threw everything at United in the second period but could find no way past Slovak stopper Dusan Pernis in inspired form.
The match went ahead following a late pitch inspection, with heavy rain putting the match in doubt before Conroy decided the Tannadice surface was playable.
The Terrors made three changes, with central defender Darren Dods stepping in for the ill Garry Kenneth, midfielder Swanson replacing the injured David Robertson and Jon Daly preferred in attack to top scorer Damian Casalinuovo, who had to settle for a place on the bench.
The visitors made one change from the side that defeated Irvine Meadow 3-0 in the Active Nations Scottish Cup last week with Patrick Cregg replacing Colin Nish, who started on the bench.
The home side made a positive start, Buaben shooting wide within the opening 30 seconds before Hibs keeper Graham Smith pulled off a terrific one-handed save to deny a deflected Paul Dixon free-kick.
Pernis had to show his agility to punch away a left-footed volley from Derek Riordan after 17 minutes as Hibs made a rare foray forward.
Smith again rescued the visitors with a fine save to keep out Andy Webster's header from an inviting cross by the lively Craig Conway after 26 minutes.
Hibernian made an early change with Abdessalam Benjelloun replacing fellow Moroccan Merouane Zemmama after 31 minutes.
A moment of controversy followed soon after when Buaben was fouled by Hogg on the edge of the area. Conroy awarded the penalty after his initial indecision and Hibs may have felt justice was done when Smith excellently saved Goodwillie's spot-kick. The United striker's misery was complete when he headed the rebound wide.
However the inevitable opener for United came three minutes later when Swanson kept his composure to slot the ball coolly past Smith after a neat passing move.
John Rankin went close to equalising with a dipping free-kick before Goodwillie, eager to redeem himself, fired just wide as the frantic half came to a close.
John Hughes made a second change at the interval, with Steven Thicot coming on for Cregg to make his first appearance of the season and the Easter Road side adjusting their formation and deploying a three-man defence.
Anthony Stokes fired wide soon after the re-start to signal Hibernian's new attacking intent following their drab first-half showing. Pernis then had to look lively to gather at the feet of Riordan after intelligent play by Benjelloun.
Liam Miller was next to try his luck for Hibs when the Irishman raced onto Ian Murray's pass but again the inspired Pernis made a splendid save.
Nish was thrown on for the visitors with 16 minutes to go at the expense of the unusually quiet Stokes.
Rankin fired wildly over the bar before goalscorer Swanson was replaced by Jennison Myrie-Williams as the Terrors fought to hang on to their lead.
David Wotherspoon was then denied by Pernis late on when his first-time volley from a searching Rankin pass was palmed round the post by the Slovakian to ensure his third clean sheet in a row.