It's now four wins on the trot for Peter Taylor's side and this win against a Darlington side, struggling at the bottom of the league, was a valuable three points towards the Tigers promotion hopes.
Darlington dominated the opening 25 minutes of the game but a combination of missed chances and an excellent performance from City's American-born goalkeeper Boaz Myhill, denied Darlington what should have been a valuable three points.
It was hard to see which side was near the top of the league and which was near the bottom as the home side kept the visitors at bay.
Myhill denied Neil Wainwright in the second minute and then saw Lee Matthews and Ryan Valentine head narrowly wide in a frantic opening 20 minutes.
In the 23rd minute, Hull counter-attacked and Elliott thought he had opened the scoring but he was denied as a raised flag caught him offside.
Two minutes later, Ben Burgess was denied by Darlington defender Matthew Clarke, but the visitors had their tails up and they took the lead after half an hour.
Darlington's defence should have cut out a Jason Price cross, but they didn't, and Elliott fired in from close range.
Darlington responded bravely and they could have equalised just before the interval, but Matthews sent a near-post header narrowly wide.
The second half was something of a non-event. City seemed happy to give Darlington plenty of possession, probably safe in the knowledge that a lack of goals has seen the North East side free fall down the league.
After the game, Darlington boss David Hodgson said: "We still believe in ourselves and one win could turn our season around."
Hull boss Peter Taylor said: "We had to defend well but we counter-attacked with pace and power and they couldn't handle it."