Craig Levein's side cemented their third place in the Scottish Premier League with a comfortable victory over a disjointed Hibernian team who were clearly low on confidence.
Following a low-key first half, Dods broke the deadlock nine minutes after the interval and when Sandaza nodded in a second the outcome was never in doubt.
The Easter Road men were left to rue a spectacular save by United keeper Lukasz Zaluska from Colin Nish's shot with the game still goalless.
Hibs were the last team to beat their Tannadice hosts, back in September, in a performance manager Levein still rates as one of the best of the season.
However, they never really looked likely to bring United's impressive run to an end.
Following Saturday's 2-1 win over Aberdeen, Levein made only one alteration to his in-form side, bringing in Roy O'Donovan for Warren Feeney up front.
In contrast, Hibs counterpart Mixu Paatelainen wielded the axe following their poor 2-1 defeat to Inverness at the weekend.
Out completely went skipper Rob Jones and the injured Steven Fletcher, whilst keeper Yves Ma-Kalambay, who had to be substituted on Saturday with an injury, Fabian Yantorno and David Van Zanten all dropped to the bench.
In their place, Paatelainen drafted in Andrew McNeil, Steven Thicot, John Rankin, Ross Chisholm and Joe Keenan.
An early goal would have done Hibs' cause the world of good and they created a half chance after only four minutes.
Derek Riordan's free-kick was deflected into Nish's path on the apex of the six-yard box but Shiels could not capitalise on the striker's backheel in a crowded goalmouth.
McNeil, in goal after replacing the injured Ma-Kalambay at half-time on Saturday, was called into action after 10 minutes as the ever-impressive Scott Robertson set up O'Donovan but the Hibs keeper blocked at his near post.
Robertson blotted his copybook in the 17th minute, however, as Willo Flood's penetrating ball into the area panicked the Hibs defence.
The former Dundee midfielder screwed his shot wide from Sandaza's cutback before Craig Conway saw his shot gathered at the second attempt by McNeil.
Hibs edged closer to a breakthrough twice in the space of a minute.
First, in the 26th minute, Nish just failed to latch on to a poor pass back from Dods, with the striker completely missing his attempt at a backheel from a tight angle as he ran away from goal.
Ross Chisholm then sent in a drive after picking up a bouncing ball 25 yards from goal but Zaluska was alert to the danger and tipped the youngster's shot over the bar.
Robertson passed up another gilt-edged chance at the other end with seven minutes to go to the interval as he mis-hit his shot from 14 yards out after Sean Dillon's effort had been blocked into his path.
United immediately went for the jugular at the beginning of the second period but the match turned when home keeper Zaluska clawed Nish's shot away from his top left-hand corner.
Two minutes later, United took the lead from the unusual source of defender Dods.
The stopper lurked with intent having stayed forward following a long throw and when Conway's left-wing cross came in he was perfectly placed to head wide of McNeil's despairing dive.
Six minutes later, United doubled their advantage as Hibs' defence fell asleep.
Flood took a free-kick on the left flank early and Sandaza stole in to head past the exposed McNeil.
O'Donovan should have put the game beyond the Edinburgh side in the 69th minute but scooped his header from Paul Dixon's cross over from only four yards out.
Morgaro Gomis also passed up an excellent opportunity to add a third with four minutes remaining but hit the side netting after racing clear on goal.
Hibs' night was summed up two minutes later as Hogg's header and Thicot's follow-up shot were blocked by Robertson on the line.