Champions Juventus consolidated third place with a 3-1 win over Genoa as Atalanta battled to a 3-2 victory over Fiorentina to stay fourth ahead of Napoli who won 2-0 at Sampdoria.
"We're starting to see the finish line," said Conte after Inter restored their 11-point lead on second-placed AC Milan who won 3-1 at Parma on Saturday.
Moroccan Achraf Hakimi came off the bench and set up Darmian to grab the winner with 13 minutes to go at the San Siro.
"It's not easy to be under constant pressure but we're getting used to it," said Conte of their bid for a first title since 2010.
"We just need to continue pedalling to the max as we approach our goal."
Cagliari goalkeeper Guglielmo Vicario, replacing Covid-19 hit Alessio Cragno, kept his 18th-placed team in the game, denying Christian Eriksen in both halves.
Romelu Lukaku set up Alexis Sanchez who thought he had broken the deadlock after quarter of an hour, but the goal was ruled out for offside.
Conte made a double substitution bringing on Hakimi for Ashley Young after 69 minutes, with Lautaro Martinez replacing Sanchez.
It paid off and minutes later as Hakimi combined with Lukaku to tee up Parma loanee Darmian to push Inter closer to a 19th Serie A crown after finishing runners-up last season.
"On a psychological level we gave an important signal today," said former Manchester United defender Darmian.
"Conte transmits the passion and the desire to win, you can see this on the pitch.
"Everyone feels they have a role to play."
Ronaldo frustrated
Juve's nine-year reign looks set to end as they sit 12 points adrift of Inter and are now battling to stay in the Champions League berths with four points separating the second and fifth-placed teams.
Serie A top scorer Cristiano Ronaldo drew a blank to add to his frustration, as the Portuguese star failed to add to his 25 goals this campaign.
"It happens to everyone to get angry," said coach Andrea Pirlo.
"Cristiano wanted to score at all costs because he's a champion who wants to stamp his mark on matches."
Dejan Kulusevski curled in the opener after four minutes in Turin.
Juve doubled their lead after 22 minutes when Federico Chiesa forced a Mattia Perin save with a Ronaldo follow-up coming off the post before Alvaro Morata sent in the rebound.
Gianluca Scamacca headed in to pull a goal back for Genoa four minutes after the break, with Marko Pjaca then missing two quick-fire chances to equalise for the visitors.
Substitute Weston McKennie sealed the win on 70 minutes for a valuable three points before next weekend's trip to Champions League rivals Atalanta.
"Unfortunately, the points lost on the road in games which seemed straightforward on paper have cost us, so we aim to improve," said Pirlo.
Atalanta held on to fourth place, staying two points ahead of Napoli, but not without a struggle despite leading 2-0 at the break thanks to a Duvan Zapata brace in Tuscany.
Zapata headed in the first off a Ruslan Malinovskyi corner after 13 minutes with the Ukrainian providing the cross for the Colombian's second five minutes before half-time.
But Dusan Vlahovic struck twice in a 10-minute spell either side of the hour to revive Fiorentina before substitute Josip Ilicic scored the winner from the penalty spot following a handball.
Napoli returned to winning ways after their midweek defeat to Juventus with goals from Fabian Ruiz 10 minutes before the break and Victor Osimhen three minutes from time.
Sixth-placed Lazio beat Hellas Verona 1-0 thanks to a Sergej Milinkovic-Savic injury-time header.
Roma sit just behind their city rivals, beating Bologna by the same scoreline after Borja Mayoral scored to end their three-match winless run.
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