Achraf Hakimi set up Darmian to grab the winner after 76 minutes at the San Siro as Inter opened up a 13-point lead on AC Milan who travel to Lazio on Monday.
“Today’s victory is not worth only three points, not even six. It is worth nine,” said Conte.
“This win is worth 95% of the Scudetto. We haven’t won it yet but we are close.”
Juventus are equal on points with AC Milan in third place, with Atalanta a further point behind before hosting Bologna later Sunday.
With five games left to play, last season’s runners-up Inter could claim their first Serie A title since 2010 as early as next weekend.
Conte’s side had been held 1-1 in their previous two games against Napoli and Spezia.
On Sunday, Lautaro Martinez sent over the bar after connecting with a Romelu Lukaku header with Hakimi missing chances in either half.
But the Moroccan full-back teed up fellow defender Darmian for the winner to rescue the three points as he had against Cagliari two weeks ago.
Verona captain Davide Faraoni thought he had scored a late equaliser from close range but the goal was ruled out for a foul on Inter goalkeeper Samir Handanovic.
The collective explosion of joy from the ‘Nerazzurri’ matched the relief of a side for whom Conte said “the ball is getting heavier and heavier”.
“For many players, this is the first time they find themselves in this situation,” said Conte who led Inter to runners-up spot in his first season last year.
“It’s been a long time since Inter have won anything,” said the former Chelsea manager, “sometimes we suffer but we need to keep our foot on the accelerator”.
Juve’s dreams of a 10th consecutive title slipped away at the hands of former coach Conte who guided the Turin club to the first three of their current run of titles.
The champions suffered in Florence with Dusan Vlahovic putting 14th-placed Fiorentina ahead from the penalty spot after half an hour after Adrien Rabiot was penalised for a handball.
Vlahovic stepped up with a well-taken Panenka past Wojciech Szczesny for his 17th goal this campaign.
“We had a very bad first half, the match had to be tackled differently because it was fundamental for the Champions League,” said coach Andrea Pirlo.
“The second half was a little better.”
Pirlo sent on Dejan Kulusevski and Alvaro Morata after the interval in place of Leonardo Bonucci and Paulo Dybala.
And Spaniard Morata pulled a goal back immediately curling in from a tight angle.
Cristiano Ronaldo missed a chance for the winner, nodding wide with six minutes to go.
Juve risk slipping out of the top four with Atalanta playing later and Napoli just three points behind before visiting lowly Torino on Monday.
Juve sporting director Fabio Paratici had insisted before the game that Pirlo would remain coach next season if the team qualify for the Champions League.
Former World Cup winner Pirlo conceded he was not satisfied after his first season on the bench at his former club.
“The expectations were different at the beginning of the season,” said Pirlo.
“I don’t think I did the job as I wanted. You learn with every match. I’m not happy and I don’t think the club is either.”
Fiorentina are still not assured of top-flight survival as they are six points above the relegation zone.