City clawed their way back from 2-0 down on the night with three goals in the last half-hour to win 3-2 at the Etihad Stadium but went out on away goals after a 3-3 aggregate result.
Their recovery was breathtaking as two goals by Sergio Aguero and a Mario Balotelli penalty put them on the verge of a remarkable comeback.
A fairytale ending was millimetres away when goalkeeper Joe Hart's header whizzed just wide in the last minute of stoppage time after his opposite number Rui Patricio tipped it past the far post.
In the end, first-half goals from Matias Fernandez and Ricky van Wolfswinkel were enough for Sporting after their 1-0 first-leg win as City were left contemplating the two very different performances they had given either side of the break.
Mancini said: "In the first half we didn't play, but I am the manager and it's not the players' fault, but mine - I probably made a mistake.
"The players did everything, and fought on the pitch. When a manager doesn't prepare well for the game, this happens.
"When I lose a game I think about why I lost and think about where I made mistakes.
"I understand we have big players but in football every game is difficult, and in the Europa League and Champions League it is more difficult, because every team plays to win.
"Sometimes we make mistakes, but the manager is the person who always prepares the game, and I probably made some mistakes for these two games."
With City now out of all cup competitions, their only hope of silverware this season remains the Premier League, with a tough match against resurgent Chelsea next up on Wednesday.
"I am proud for the second half because we had a fantastic 45 minutes, but I am very sorry for the players and our supporters because they were fantastic in that second half," Mancini said.
"But I am disappointed in myself, not the players.
"For the second time this year, I hope I don't make mistakes against Chelsea."
He added: "If we have the same spirit as we had in the second half we can win the next game against Chelsea.
"And I am confident for the Premier League because we stayed on top for seven months and now we are in second position, one point behind. I am very confident we can win the title.
"I think that now it will be a good fight until the end of the season because we have only the Premier League and we can put all of our strength into this. I think it will be good."
Mancini had been left angrily shaking his head when Balotelli, the villain of City's departure at the same stage last season when he was sent off against Dynamo Kiev, needlessly gave away the free-kick that led to Sporting's opener.
But while Mancini said he was disappointed at yet another example of the Italian striker's ill discipline, he did not blame him for the defeat this time.
"It is not his fault we lost this game, I'm the manager, it's my fault," he said.