Making their first appearance in the FA Cup third round for nine years, the Shrews were dominant in the first half against their more illustrious opponents.
However, they were made to pay the price for missing some excellent opportunities - the best of which fell to Lionel Ainsworth - as Boro took the lead just before the break.
Emnes was the scorer, profiting on some fine work from Malaury Martin to grab his 11th goal of the season.
Much like the first half, Shrewsbury had their chances in the second period, with Terry Gornell spurning the best of them in the 74th minute.
And, despite missing openings of their own, Middlesbrough emerged victorious thanks to an industrious display.
Perhaps predictably, it was the hosts who enjoyed the better of the opening exchanges although the Shrews created the first opportunity.
Taking advantage of some poor defending, Mark Wright surged forward and struck inches wide of goalkeeper Jason Steele's post.
That spurred the visitors on, with Gornell forcing Steele into a save before Martin hit narrowly wide at the other end.
The game entered a lull thereafter, but Shrewsbury spurned the day's best chance in the 30th minute as Ainsworth was wide of the mark with the goal gaping.
The Shrews were continuing to enjoy the majority of possession as the first half wore on, with their patient approach causing the home side problems.
But Middlesbrough took an undeserved lead on the stroke of half-time through Emnes.
Martin sent a ball towards the back post and the Dutchman showed neat control to bring the ball down before slotting past a helpless Chris Neal.
Picking up where they left off, Boro looked a different side immediately after the interval, Bart Ogbeche forcing a save from Neal with a fine volley.
Despite dominating possession though, they were struggling to fashion any real openings, with a high-and-wide Martin free-kick summing up their efforts.
The Shrews were soon on hand to remind the hosts of their presence as Marvin Morgan missed the target prior to Matt Richards forcing a save from Steele.
Emnes could have finished the game off soon after, failing to reap rewards from his own good work as he hit just wide.
But it was the Shrews who were continuing to create the better chances, with Wright striking narrowly wide before Gornell somehow repeated the trick from close range.
Morgan forced a fine save from substitute keeper Danny Coyne in the dying seconds as Boro held on.