Achraf Hakimi set up Darmian to grab the winner on 76 minutes at the San Siro.
Inter are 11 points ahead of Atalanta, who moved second with a 5-0 rout of 10-man Bologna, with five games remaining.
AC Milan are a further two points behind in third before they travel to fellow top-four hopefuls Lazio on Monday.
“Today’s victory is not worth only three points, not even six. It is worth nine,” said Conte.
“We’re 95% of the way towards the Scudetto. We haven’t won it yet but we are close.”
Juventus are level on points with AC Milan but drop to fourth place after being held 1-1 at Fiorentina.
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 this was a crucial game 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, failing to score for the third consecutive game.
Juve risk slipping out of the top four, with Napoli just three points behind before visiting lowly Torino on Monday.
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.”
In Bergamo, Atalanta kept their push for Champions League football on track with an impressive display from striker Luis Muriel.
The Colombian’s back-heel flick set up Ruslan Malinovskyi for the opener on 22 minutes, adding a second himself from the penalty spot just before the break for his 19th goal this campaign.
Jerdy Schouten’s sending off four minutes after half-time for a dangerous tackle on Cristian Romero complicated the task even further for 12th-placed Bologna.
And Remo Freuler added a third on 57 minutes, with Malinovskyi setting up Duvan Zapata for the fourth before the hour. Substitute Aleksei Miranchuk completed the rout with 17 minutes to go.
Seventh-placed Roma lost 3-2 at Cagliari before their Europa League trip to Manchester United, as the Sardinians, now 17th, boosted their hopes of top-flight survival.