Hayden Hackney's goal was just his second of the season in all competitions
Hayden Hackney scored the only goal, opening up his body to slide home an Isaiah Jones cross in the first half.
Cole Palmer missed three presentable opportunities as Chelsea, who had 18 shots, but only five on target, failed to score for the sixth time this season.
"It is difficult to talk about our performance," said Chelsea boss Mauricio Pochettino.
"We created some chances, but we were not clinical enough. This first half of the season, we have created some chances, but we have not won because we were not clinical enough - we must keep pushing."
Chelsea, who sit 10th, have missed 40 big chances - as defined by statistical analysts Opta - in the Premier League so far this season, the second-highest total in the top flight, behind Liverpool (41).
They are looking to bring in a forward in the next two transfer windows but it is unlikely they will be spending the huge money needed to land Brentford's Ivan Toney or Victor Osimhen from Napoli.
Boro, who have lost six games at home in the Championship this season, can now dream of the reaching the final, 20 years on from winning the tournament.
They came into the came with 12 first-team players unavailable and lost striker Emmanuel Latte Lath and left wing-back Alex Bangura to injuries in the opening 20 minutes.
"The way they stuck together was very special and makes me proud to see that," Middlesbrough manager Michael Carrick told Sky Sports.
"We had to show a bit of everything and I'm buzzing. I know what it means to everyone, it was fantastic for me to see so many people so happy and it's pretty special.
"We will be ready for the second leg and attack the game and see what happens. Football is about dreaming and things happen like tonight which probably shouldn't happen. We go there with our eyes wide open and know what to expect, but it won't stop us trying."
The second leg takes place on Tuesday, 23 January at 19:45 GMT.
Liverpool face Fulham in the other semi-final, with the final taking place at Wembley on Sunday, 25 February.
-- Courtesy of BBC Sport