The Turin club saw their 12-match winning league run halted after being held 1-1 at Stadio Olimpico on Saturday, but it could have been worse for Juve, who had Gianluigi Buffon sent off for bringing down Miroslav Klose when he was through on goal.
Antonio Candreva added further punishment to the reigning champions by sending substitute goalkeeper Marco Storari the wrong way from the spot.
But Juventus bounced back, deservedly equalising on the hour when Stephan Lichtsteiner's teasing cross was looped into the far corner by the head of Fernando Llorente.
The result means that Juve extended their unbeaten league run against Lazio to 19 matches, and Conte was thrilled with the way his side had responded to the setback of losing their goalkeeper and captain.
"We wanted to win, both with 11 men and with 10," he told Sky Sport Italia.
"What we try to drill into the players is a winning mentality and these lads know full well every game must be played to win, even when the goalkeeper is sent off after 25 minutes.
"I think that in difficult times the men emerge as well as the players, which is what happened today.
"We picked up a point and I think the team proved again tonight why they are top of the table.
"Juventus never gave in, continued to play football and tried to win the game despite going down to 10 men.
"It's a shame we interrupted the 12 consecutive Serie A victories, but these things happen and Lazio deserve credit for their performance."
Conte was also disappointed to see Buffon sent off, as Juve were already punished by the awarding of a penalty, although he conceded that the referee was acting according to the rules.
He added: "Inevitably, it's disappointing that there's this rule, because I think a penalty is punishment enough without the need for a red card.
"I've said that for years, regardless of whether it's in our favour or against, as it ruins the game, especially as sometimes they give a yellow card, sometimes a red, so it's never quite clear."
