The defending champions held a two-goal lead and looked comfortable but United refused to lie down and made it an uneasy night for Gordon Strachan's side.
Celtic seemed to suffer from a hangover from their weekend defeat at Ibrox as Dundee United seized the initiative in the early stages.
Morgaro Gomis set up a long-range effort for Craig Conway and the midfielder's shot whistled narrowly wide.
But a burst forward from Scott Brown after 15 minutes brought the Celtic fans to life and he set up Georgios Samaras whose shot cracked against the left-hand post with Lukasz Zaluska beaten.
However, United refused to be knocked off their stride and it took a magnificent save from Artur Boruc to keep out a netbound header from Francisco Sandaza.
This time Celtic heeded the warning and scored within a minute. Garry Kenneth conceded a corner and Shunsuke Nakamura's flag-kick was met by Glenn Loovens and his header found the net.
United claimed that one of their players had been fouled at the corner but referee Stuart Dougal was not interested.
The goal breathed new life into Celtic and Aiden McGeady - controversially left out against Rangers on Saturday - came close with a shot which was just a yard too high.
Then, Scott McDonald curled a shot wide as Celtic piled on the pressure in search of a second goal before half-time.
The home side won a series of corners shortly before the break but United held firm. although Lee Wilkie was fortunate to escape when a cross from Samaras came off him and flashed just inches wide.
Sandaza had a chance early in the second half from a tight angle but Boruc was alert to the situation and managed to get a foot to the ball.
Kenneth then had an opportunity in another promising break but he failed to make proper contract eight yards out and Celtic survived.
But it was Celtic who extended their lead after 52 minutes. McDonald and Brown combined to set up Samaras and he sent a right-foot shot from 20 yards into the corner of the net.
Kenneth then failed to deal with a Lee Naylor cross and the ball broke for Brown but his diving header was straight at Zaluska.
But Celtic were stunned after 57 minutes when a superb cross from Paul Dixon found its way to David Robertson and he made no mistake.
Moments later, substitute Andis Shala was just too high with a header in front of goal from a Robertson cross as United enjoyed their best spell.
Prince Buaben had a couple of late chances, but Celtic held on for the points.