Deila rested most of his team following Wednesday's 3-2 victory over Malmo, with the second leg of the Champions League play-off to come on Tuesday in Sweden.
Celtic lost 1-0 at Inverness at this stage last year after Delia made 10 changes in between their unsuccessful qualifiers against Maribor, but a United side fresh from a 4-0 drubbing at Hamilton offered little threat.
Leigh Griffiths opened the scoring in the 17th minute at Tannadice and Celtic should have been out of sight before half-time.
An own goal from Mark Durnan a minute before the break appeared to have killed the game off but Chris Erskine pulled one back from the spot with the last kick of the half.
The second period was a tighter affair but Callum McGregor sealed three Ladbroke Premiership points with a deflected strike in the 74th minute.
Young defenders Saidy Janko and Eoghan O'Connell were among those coming in but Delia still fielded an experienced side with eight internationals.
It was the only two players to retain their places who combined for the opener as Emilio Izaguirre's ball found its way through for Griffiths, who rounded Luis Zwick and rolled the ball home.
United claimed offside but Durnan could have removed any debate by making a better attempt to cut the pass out.
The goal was no surprise as Celtic had dominated the opening stages and Durnan had made an excellent goal-line clearance from Kris Commons before Charlie Mulgrew curled a free-kick just over.
Celtic continued to carve out chances after the opener
Commons alone had four more decent penalty-box opportunities in the first half and forced three good saves from Zwick.
Commons also set up Griffiths on the break but Zwick saved with his foot and Celtic peppered several long-range shots not far wide.
In among the wave of Celtic pressure, United had a strong penalty claim rejected midway through the half when goalkeeper Logan Bailly recklessly went through the back of Robbie Muirhead after being caught in possession by the United forward
Referee Willie Collum waved play on.
The hosts should have levelled with their first decent attack five minutes before the break when good skill from Blair Spittal set up Ryan McGowan but the unmarked midfielder shot wide from 18 yards.
United kept the pressure on but were caught on the break when Griffiths released Commons and Durnan knocked the ball past Zwick as he cut out the midfielder's return pass.
But United immediately got back in the game after Efe Ambrose tripped Scott Fraser as the United midfielder ran towards goal and Erskine sent Bailly the wrong way from the spot.
Ryan Dow came on for the injured Fraser early in the second half for his first appearance since being hit by hamstring problems in March.
The creative midfielder could not spark United into life, although they looked a lot more solid in the second half.
Scott Allan came on for his Celtic debut on the left of midfield midway through the half following his move from Hibernian and almost notched within seconds, but Zwick caught his chip following a poor clearance from the goalkeeper.
The Celtic fans regaled the former Rangers fan and target with songs about his shift in allegiances, but it was another sub who stole the limelight when McGregor's 20-yard strike spun past Zwick after deflecting off Coll Donaldson.
United refused to fold and Bailly made a decent stop from Erskine before theatrically stopping two free-kicks from Aidan Connolly.
Source : PA