Beech’s League Two side were paired with the Blades at the first-round stage after their relegation to the Championship last season, after two campaigns in the top flight.
The Cumbrians will make their first trip to Bramall Lane since early 2014 on August 10, days after United open their new campaign at home to Birmingham City.
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And Beech, the former Blackpool and Hartlepool United midfielder, said: “That’s amazing, really – we’ve got the best team in the competition at this level.
“They’ve just been relegated and I look at the past, in terms of their team, managed by a manager who I greatly respect in Chris Wilder.
“He’s been unbelievable – he’s done it the hard way, the long way, even though he’s not managing them now.
“He hasn’t got a foreign-sounding name, Chris Wilder, so he’s had to crack on and take big knocks along the way, so he probably is in the sun somewhere right now.
“We’re playing his old team. I know a lot of people at Sheff United, so it’s one to look forward to.”
United’s last game against Carlisle, back in League One, featured THREE players who were named in Gareth Southgate’s England squad for this summer’s European Championships – goalkeeper Jordan Pickford, who was on loan at Carlisle from Sunderland, Conor Coady and Harry Maguire.