Evans fired home deep into added time past Andy Warrington and in off the post as the Blades edged through, when penalties looked likely to decide this second round Northern tie.
Chris Porter had put the visitors infront early on in a bitty first half, as he outpaced Michael Raynes on the left before shooting low past Warrington.
Danny Harrison and Raynes himself went close at the other end before the break.
The Millers stepped up the pace after half time with Lewis Grabban and Alex Revell looking threatening despite Evans striking the crossbar.
Both men combined to tee up Ben Pringle at the back post but the midfielder's shot was blocked by Nick Montgomery.
The equaliser eventually came when Jason Taylor fed Grabban who's low shot struck the foot of the post and Revell smashed in the rebound.
Danny Schofield and Pringle then wasted good chances to put the Millers infront after a succession of corners at the other end and a fine stop by Warrington to deny Mendez-Laing.
And after injury to Mark Bradley with time ticking down, Evans struck the hammer blow in the final minute of stoppage time to send the Blades through.