But Chris Wilder's side made hard work of securing their first home win since before Christmas.
After a dire first half, Oli Johnson headed the U's in front six minutes after half-time.
But after Scott Rendell had missed a glorious opportunity to double the lead, the Daggers equalised when Mark Arber was left unmarked at the far post.
The visitors almost went ahead when Adam Cunnington hit the outside of a post, but Constable went through with ten minutes left to drill home the winner.
When Johnson scored six minutes after the break you thought Oxford would go on and take the three points against their relegation-threatened opponents.
But they were forced to do things the hard way.
Liam Davis started the move for Johnson's goal.
The left back surged forward and was shown down the line by Spillane.
He had no hesitation in accepting, going past the defender and putting in a great ball that was met by the head of Johnson who scored on his first start.
However, Oxford were again made to pay for failing to get the second goal when Arber levelled 18 minutes from time.
Davis needlessly gave the ball away, and was saved by a superb tackle from Jake Wright, who timed his slide to perfection to keep out Nurse's attempt on goal.
From the corner, however, a cross from substitue Medy Elito - who had only just come on - found Arber in acres of space and his header gave Clarke no chance.
Wilder had sent on Swindon loanee Mehdi Kerrouche with 11 minutes remaining, and he played a big part in the goal.
Kerrouche held the ball up well, Johnson played it on and Constable fired home at the near post.
Both sides had been without a game for more than a fortnight - and with four new faces in the U's squad, the early stages were poor.
Johnson lined up on the right of the three-man front line, Rendell in the middle, with Constable pushed out to the left.
Kerrouche was on the bench, alongside Lee Holmes, who was only signed earlier that day on loan from Southampton.
Mark Wilson made his debut in midfield, with Asa Hall replacing Simon Heslop, who was not in the 16 after picking up a heel injury.
The opening 25 minutes was littered with misplaced passes and from Oxford's point of view, new players trying to find their feet.
The only noteworthy incident in that period was a backpass from Scott Doe that sold keeper Chris Lewington short, and he had to be quick to clear the danger from the onrushing Constable.
United should have gone ahead in the 26th minute however.
Some great play from Rendell saw him play in the ball behind the Dagenham defence, and Asa Hall ghosted in, headed down, but then shot wide when under pressure.
He should have done better.
Rendell's header from a free-kick was then saved, before at the other end Micky Spillane shot high and wide from the edge of the box when United's defence were slow to close him down.
But after that flurry of activity - all five minutes of it - we went back to the misplaced passes and a lack of creativity.
The first time a goalkeeper was tested came in the 42nd minute, when Ryan Clarke had to get down well to keep out Billy Bingham's 20-yard effort.
The half-time whistle brought a sprinkling of boos, and Wilder made his feelings known by sending out the players early for the second half.
And he also changed the system, bringing on Holmes for Hall and switching to a 4-4-2 formation.
United started the second half well, winning two quick corners.
And from a Davis's surge, they took the lead thanks to Johnson's fine header.
They had a scare in the 58th minute, however, when Arber's header from a corner was cleared off his own line by Constable.
Oxford then had a great chance to double their lead when from Johnson's terrific ball over the top, Rendell was sent clear.
With the Dagenham defence looking for offside, the loanee bore down on goal, cut inside, but over-ran the ball and Lewington gathered.
It was a golden opportunity wasted.
Constable then shot high and wide from the edge of the box as United turned up the pressure - but they needed the second goal.
Holmes looked lively down the right, and put in a number of good crosses.
Clarke had to once again be alert when Jon Nurse was left unmarked at the far post, but United's keeper easily kept out his header.
Arber's goal looked as though it might deny United victory, before Cunnington's strike beat Clarke but rebounded off a post and behind.
But Constable had the final word, his strike giving United a vital three points that keeps them in that final play-off place.
Oxford Utd: 4-3-3 - Clarke; Whing, Duberry, Wright, Davis; Hall (Holmes 46), Wilson, Leven; Johnson (Tonkin 83), Rendell (Kerrouche 79), Constable. Subs: Worley, McLaren.
Dagenham: 4-5-1 -: Lewington; Spillane, Doe, Arber, Ilesanmi; Nurse, Ogogo, Saunders, Bingham (Cunnington 56), Green (Elito 68); Scott (Reed 85). Subs:Hogan; Rose.
Attendance: 5,653 (119 from Dagenham)