The Blues arrived at the Championship side as the favourites to win on Tuesday night but a lack of cutting edge cost them once again.
Mauricio Pochettino has ensured his side have taken the Carabao Cup seriously this season, with wins coming against fellow Premier League sides Brighton and Newcastle.
Chelsea also claimed victories against AFC Wimbledon and Blackburn Rovers, but Middlesbrough have proven to be their toughest opposition.
Cole Palmer, who has been excellent for the Blues since his summer arrival, had two glaring chances to open the scoring.
The first saw him drag a golden opportunity wide of the post after Jonny Howson gave away possession inside his own half.
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Palmer's second missed chance was the worst of the night, with the 21-year-old putting the ball over from around five yards out.
This came just moments after Hayden Hackney gave Middlesbrough the lead, a low finish inside the area from a well-worked attack.
Chelsea went into the break the losing side for just the second time this season, with the other occasion their quarter-final win over Newcastle.
They came from behind to draw late on but this wasn't to be repeated at the Riverside Stadium.
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Chelsea came out the better side in the opening stages of the second-half but their problems in attack were on show yet again.
The Blues dominated possession but had nothing to show for it in the final third, with frustration beginning to creep in.
Middlesbrough were more than happy to soak up the pressure and hit their Premier League opponents on the counter-attack.
Their second-half efforts going forward were also disappointing but it's Michael Carrick's side who hold a 1-0 aggregate lead.
Chelsea will host them in the second leg at Stamford Bridge on January 23, with a much improved performance needed if they are to overcome the deficit.
If there is any positive to take from the first leg, it's that they did not concede a second on what proved to be another dismal night.