Inter were staring down the barrel of a sixth defeat of the Serie A season after a Felipe Anderson double put Lazio into a 2-0 half-time lead at San Siro.
But goals from Mateo Kovacic and Rodrigo Palacio after the break earned Mancini's side a 2-2 draw.
Centre-back Juan conceded that Inter were far below the required standard in the first half and credited head coach Mancini for their turnaround.
"We approached the game in the wrong way in the first half but we turned things around after the break," he told Inter's official website.
"A lot of that was down to what the coach said to us at the interval. We need our second half approach for the full 90 minutes in matches and to consistently do the right things."
A point means Inter head into Serie A's mid-season break in 11th place and six points adrift of Lazio - who occupy the third and final UEFA Champions League position.
Full-back Danilo D'Ambrosio is confident that Inter have more than enough time to improve their league standing in the new year.
"We put in a great second-half performance and showed character to come back," he said.
"We still have a lot of matches to play and there's plenty of time to move back up the table. We'll keep going."
