Elokobi, who was captain for the day, equalised in the second half through a headed goal but Nicky Ajose's double meant the 10-man Crewe Alexandra left SportsDirect.com Park with the three points.
George Ray had earlier seen red for the Railwaymen for a professional foul on Jonathan Forte but the result leaves Latics three points adrift of the play-off positions.
Latics started the game brightly and went close in the opening minute through Mike Jones.
The midfielder cut inside from the left and unleashed a shot which was deflected away from danger.
Carl Winchester had Latics' next chance on six minutes as he nicked the ball in midfield, drove at the defence but saw his shot well held by Crewe goalkeeper Ben Garratt.
Jabo Ibehre was proving to be a constant threat in attack and almost put Latics ahead two minutes later.
The ball fell for the striker perfectly on the edge of the box and he instinctively shot for the top corner but it fell inches wide.
Latics continued to dominate possession but found goalkeeper Garratt in good form.
Midway through the first half, Garratt misjudged the flight of the ball and Ibehre almost capitalised on his error but the ball was cleared away from danger by the defender.
The home side were almost caught on the counter attack minutes later as Ajose lobbed Paul Rachubka but the goalkeeper plucked the ball out the air and tipped it over the bar.
From the resulting corner, Joseph Mills was forced to hack the ball off the line as Crewe knocked on the door.
Captain for the day Elokobi rose highest from a corner five minutes before the break but Garratt saved his header well.
Seconds before the end of the half, Ibehre thought he had given Latics the lead as he wriggled past three defenders but saw his shot saved low down by Garratt.
Latics started the second half much as they started the first with waves of attacks.
Full backs Brian Wilson and Mills were providing much width to Latics forward play and it started to become effective after the break.
Ten minutes into the half Crewe were reduced to ten men as George Ray pulled back Jonathan Forte as he raced through on goal and the referee deemed him to be the last man.
From the resulting free kick, Jones smashed his 20 yard free kick narrowly past the post as Latics looked for an opening goal.
The home side were creating chance after chance and Mills was the next to test the goalkeeper but his shot from 25 yards crashed against the crossbar.
For all the home side's domination it was 10-man Crewe that took the lead against the run of play from the penalty spot.
Elokobi was adjudged to have fouled a Railwayman in the box and Ajose stepped up to put the ball beyond Rachubka to give the visitors an unlikely lead.
That lead lasted all of six minutes as Elokobi made up for his penalty mishap to head Latics back into the game.
Tidser's corner found the head of the rising defender and he knocked it home from six yards.
Latics were doing all of the pressing and looked the more likely to snatch the winning goal as substitute Amari Morgan-Smith almost latched on to a cross to the back post but the ball exceeded him.
Heading into the final five minutes of the game, Ibehre thought he had knocked the winner for Latics but his header fell agonisingly wide of the goal.
Completely against the run of play, Crewe took the lead for a second time on 86 minutes.
Ajose was allowed acres of space inside Latics' half and chipped goalkeeper Rachubka from 30 yards to win the game for Crewe Alexandra.
Latics pressed in the final minutes of the game and maybe should have been awarded a penalty as Ibehre was tugged back in the area but the referee waved away the late protests.
But for all Latics' possession and dominance, Crewe went away with all three points on the Boxing Day encounter.
Oldham Athletic: Rachubka, B.Wilson, J.Wilson, Elokobi, Mills, Jones, Winchester (Dayton 84'), Tidser, Forte (Morgan-Smith 74'), Philliskirk (Johnson 74'), Ibehre.
Subs not used: Coleman, Dieng, Poleon, Mellor.
Crewe Alexandra: Garratt, Guthrie (Cooper 46'), Davis, Ray, Turton (Stewart 60'), Bailey, Leigh, Ness, Ikpeazu (Jones 80'), Ajose Grant.
Subs not used: Shearer, Tootle, Nolan, Haber.
Goals: Oldham Athletic (Elokobi 75') 1-2 Crewe Alexandra (Ajose 69', 86')
Attendance: 4,782 (564 visiting fans)