Substitute Marcus Haber had squared the Capital One Cup game for Crewe in the seventh minute of added time after Beckford had struck in the first minute of stoppage time.
But the experienced marksman, who was scoring his first goals of the season, flicked the ball under goalkeeper Ben Garratt in the 106th-minute to secure Wanderers their 3-2 victory.
It was harsh on Crewe and Brad Inman, in particular, who tore at an under-strength Bolton defence with a high octane display, which included hitting a superb second-minute opener.
Inman cut in from the left flank, passed young Wanderers full-back Oscar Threlkeld and unleashed a screaming right-footer out of reach of Andy Lonergan.
The stand-in Bolton keeper prevented his side falling further behind by palming around Chris Atkinson’s long-range effort.
Bolton were on the backfoot for much of the first period as a side showing nine changes from the one which went down at Brighton at the weekend were far from comfortable with Crewe’s energy and movement.
Lonergan parried another Inman effort into the path of Vadaine Oliver who could only haplessly volley wide of a gaping goal.
Wanderers levelled against the run of play when Darren Pratley rose highest to greet Liam Feeney’s cross with a downward header five minutes before the break.
Crewe survived penalty appeals when substitute Lee Chung-yong appeared to be tripped by Matt Tootle in the closing stages.
And Beckford appeared to have grabbed the winner for Wanderers when he drove in after Matt Mills flicked on a Feeney cross.
But Crewe substitute Haber was on hand to head home at the far post to force extra time.