The full-back was due to see his previous deal expire at the end of the season, although it was widely expected fresh terms would be agreed.
Inter announced the news on Thursday, less than 48 hours after Darmian scored the winning goal in the Coppa Italia quarter-final against Atalanta.
"Hi Inter fans. I'm delighted to keep on wearing these colours. See you at the stadium," Darmian said in a video posted on social media.
Darmian joined Inter in 2020 from Manchester United, initially on loan before making a permanent move the following year.
He was part of the Serie A winning side in his first year at the club and lifted the Coppa Italia last season, as well as the Supercoppa Italiana – a trophy Inter retained last month by beating rivals Milan.
Belgium international Lukaku set them on their way to a fifth consecutive top-flight win with a goal inside the opening minute.
Antonio Conte's side largely dominated proceedings thereafter, with Nicolo Barella striking the crossbar, and they wrapped up all three points in the second period through Darmian and Sanchez.
The result moved them seven points ahead of second-placed Milan, although Stefano Pioli's side can rein them back in with victory over Roma later on Sunday.
Inter started in blistering fashion, Lukaku firing into Mattia Perin's bottom-right corner from just outside the penalty area after 32 seconds – Inter's second-fastest Serie A goal since Opta started tracking such data in 2004-05.
Darmian clipped wide after 14 minutes, while Barella hit the bar with a long-range effort soon after.
Perin then kept out Lukaku's powerful strike shortly after the half-hour mark as Inter ultimately failed to double their advantage before the interval.
The hosts put the result beyond doubt in the 69th minute when Darmian lashed home his first Serie A goal for the club from 12 yards after being picked out by Lukaku.
Substitute Sanchez then added gloss to the scoreline 13 minutes from full-time, heading in from close range after Lukaku's effort had been repelled by Perin.
