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After a 2-2 draw at Anfield, Liverpool prevailed 14-13 in a shootout in which six players scored twice, to reach the fourth round.
"Our players showed great resilience to keep going," said Rodgers.
"And at the end, when it was needed, they showed wonderful composure with some of our young players stepping up."
Liverpool took the lead after just 10 minutes through debutant midfielder Jordan Rossiter, 17, but Boro equalised when Adam Reach headed in after the break.
Suso drilled home for the Reds in extra time but Patrick Bamford's last-minute spot-kick, after Raheem Sterling gave the ball away and Kolo Toure brought down Bamford, took the tie to penalties.
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The shootout featured 20 successful penalties in a row and equalled the record scoreline in English professional football, set in the Football League Trophy in August 2011, when Dagenham & Redbridge beat Leyton Orient.
"It's a relief to get through," added Rodgers. "We are disappointed with the goals we conceded. We are making too many mistakes at the moment. But what this team has is character and we needed that.
"We were practising penalties last night. We are humble enough as a group to think we would maybe go to penalties because Middlesbrough are a good side and are doing very well, so everyone practised."
The match was a triumph for Liverpool's young players, with Rossiter scoring the opening goal on his debut and 18-year-old Jordan Williams finding the net in the shootout on his first appearance.
Suso, a 20-year-old Spanish forward, scored his first Liverpool goal and netted two penalties before Boro winger Albert Adomah pushed his effort wide to finally bring the cup tie to an end.
"It was a good night for our young players," added Rodgers. "Young Jordan Rossiter at 17 years of age - he could walk to Anfield - he lives just around the corner, and I was delighted for him.
"He got the opportunity and showed wonderful composure to get that finish.
"I was also impressed with young Jordan Williams - he came on and showed real good stature and composure in the game. He is a wonderful footballer and a young player who is developing well.
"And young Suso hasn't had a lot of game time at all, but he has kept working and I thought he was really bright for us when he came on, and took his goal really well."
Liverpool boss Brendan Rodgers praised his young players' composure as the Reds beat Middlesbrough in the longest League Cup penalty shootout.