They battled bravely but were always behind once the visitors had taken the lead courtesy of a Marc Wright try after a hard fought first 20 minutes.
Brett Marriott's conversion increased the lead and then Joe Langford crossed for 'Bury, with Marriott again converting to stretch the advantage.
Elland replied on 35 minutes with a Gareth Webb try which he converted himself, but their revival hopes were short lived as eight minutes later Marriott crossed the whitewash and added the extras to make the half time score 18-6.
Bury carried on where they had left off after the break with Stephen Eyles grabbing a try which Marriott again converted.
Danny Murgatroyd pulled the home side back into it with a try he converted himself, but five minutes later Luke Brearcliffe crossed to put the visitors back in control.
Stephen Pryce crossed for the home side in the closing minutes, but it was too little too late to save them from defeat and the drop.

Greetland AR 20 v Crigglestone AB 34
GREETLAND fought back superbly against the unbeaten Division One leaders after looking set for a hiding at half time.
They trailed 28-6, their points coming from a try by loose forward Andy Wickings, who sold an outrageous dummy to go under the posts, and Alan Hindle conversion.
Rock solid defence and some sparkling open play enabled them to cut the gap to eight poins with tries from Keleen Smith, Hindle and Wickings plus a Hindle conversion.
However, the Wakefield side had the last say to seal promotion.

Emley Moor 44 v Boothtown Terriers 20
BOOTHTOWN showed lots of character in a battling show against a big Emley side in Division Two.
As snow showers whipped across the ground, Emley started strongly and opened up a 12 point lead inside 10 minutes.
But the Terriers got in the game with Ryan Barnes and Paul Rawson leading from the front and Andy Charlesworth putting in a spectacular flying tackle.
Midway through the half good work by Barnes and Dean Robertshaw led to Ben Mallas spinning in the tackle to score, with Rawson converting.
Two more Emley tries followed before Andy Charlesworth fed son Luke who dived in at the corner for a well worked try which made the half time score 22-10.
Emley increased the lead with a penalty and a try but Boothtown wouldn't lie down and after a break from Luke Brown, Barnes threw off the defenders to score.
Then came the try of the match as with seemingly nothing on, Jason Dodd took the ball 70 yards from the try line, took on the defence and broke clear. After side-stepping the fullback and handing off the chasers, he made it to the line for a super score.
That brought Boothtown to within 10 points but two further tries sealed a home win as the game ended in a blizzard.


Queensbury A 12 v Underbank 22
A YOUNG `Bury team pushed experienced foes all the way in Division Four before eventually falling to a narrow defeat.
The snow thawed just in time for the game to take place, and though Underbank opened the scoring, a try from 17-year-old Luke Cheney brought `Bury level.
The visitors scored again but 'Bury hit back with a try from Kelvin Daulat to make the half time score 8-8.
Underbank took the lead for a third time but a second try from Daulat levelled it once more.
'Bury, for whom Jimmy Wright and Aaron Moore were outstanding, continued to test the Underbank defence but two late tries gave the visitors the points.