The Premier League side dominated at the Riverside Stadium but squandered a host of chances, with Cole Palmer particularly wasteful.
Hayden Hackney's first-half goal earned the Sky Bet Championship hosts a slender advantage ahead of the return meeting at Stamford Bridge in a fortnight's time.
'We had too many chances to score where we didn't score - that is football,' Blues boss Pochettino told Sky Sports.
'Of course we are disappointed. It is the first half of the tie. We have another 90 minutes at Stamford Bridge and we have to be positive.
'We made some mistakes and we were punished for that.'
Some of Chelsea's travelling fans reacted angrily to the defeat at full-time.
'If we assess the performance, overall we were the better side, we create more chances, we have clear chances,' continued Pochettino.
'But we didn't score and we were not clinical and that has happened a lot this season - so many games we didn't win because we weren't clinical enough.'
Boro boss Michael Carrick said: 'I thought the lads were exceptional, outstanding - effort, spirit was just off the scale, I'm just so, so proud of them. It was an incredible night.
'Football's about dreaming, things happen in cup competitions. Tonight shouldn't probably have happened and we found a way to make it happen - the boys were incredible.
'We'll go down there with our eyes wide open and know what to expect but it doesn't stop us from trying.'
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