After the break, Vokes tried his luck from distance, only for his shot to go wide, but just past the hour the Wales international got on the end of Edwards' right-wing cross to head an equaliser into the bottom corner.
Jarvis missed a golden chance to put Wolves ahead after Brad Jones had tipped Andrew Keogh's cross into his path, and four minutes later Boro struck the decisive blow.
Downing's great run set up Emnes, whose low left-foot shot found the back of the net off the post and Hennessey to send Boro into the last 16.
Wolves manager Mick McCarthy on his decision to make five changes:
"I think if our aim is to go to Reading and get something, then to have all my team play 90 minutes ahead of playing the most dynamic team in the league - and arguably the best team in the league - would be folly on my part.
"We still tried to win the game. We had three strikers on trying to win the game because we didn't want a replay. But it was not to be.
"We will now play a league game instead of a cup game and that's it. It's all about the league now and we are in a good position."
Middlesbrough manager Gareth Southgate:
"I think it is important to win because in the last two years we have made the quarter-finals twice, and it has coincided with our league form improving.
"A lot of people expected us to get turned over and that it would be hard for us to get a result because they have got momentum and a league position and we haven't got that at the moment. But that didn't happen.
"It was a big ask as we had a young side out there but they responded to everything that was thrown at them and that is great credit to the group of players."
Wolverhampton: Hennessey, Edwards, Shackell, Collins, Hill, Jarvis (Ebanks-Blake 81), Henry, David Jones, Reid (Kightly 65), Iwelumo (Keogh 52), Vokes.
Subs Not Used: Craddock, Friend, Ikeme, Foley.
Booked: David Jones.
Goals: Vokes 63.
Middlesbrough: Jones, Bates, Wheater, Riggott, Taylor (McMahon 7), O'Neil, Sanli, Shawky (Hoyte 46), Adam Johnson, Downing, Alves (Emnes 70).
Subs Not Used: Porritt, Bennett, Turnbull, Craddock.
Goals: Alves 44, Emnes 83.
Ref: Rob Styles (Hampshire).
BBC Sport Player Rater man of the match: Middlesbrough's Tuncay Sanli 7.79 (on 90 minutes).
Please note that you can still give the players marks out of 10 on BBC Sport's Player Rater after the match has finished.
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