The pair have both put pen to paper to extend their current deals, which will now see them remain at Hove until at least the end of the 2016 season.
Crawley-born wicketkeeper-batsman Brown, 26, made his first-class debut in 2007 and has gone on to make nearly 3,000 runs in 67 matches after progressing through the Sussex youth and academy set-up.
He made his first-class best of 163 during the LV=County Championship match with Durham last summer – the highest-score ever by a Sussex number seven as he shared an incredible record sixth-wicket partnership of 335 with Luke Wright.
Brown said: “I’m really pleased to have signed again for the club I have supported as a kid and, hopefully, over the next two years I can help the squad add to the rich history of the club and get back to winning trophies.
“That’s the priority now as we have been close in four-day cricket but now we need to kick on as a squad and challenge on all fronts. I’m looking forward to being a part of that big challenge over the next two years and hopefully beyond.”
All-rounder Finch, 20, signed a full professional contract following the 2013 season and the Hastings-born product of Sussex’s age group and academy system has experience with the England under-19’s setup.
He made his T20 debut for Sussex in 2014 against Sri Lanka and also came to the fore in the final Royal London One-Day Cup outing of the season, when he smashed an unbeaten 92 from 58 deliveries against Glamorgan at Hove.
Finch said: “I’m honoured to have been offered another year. I’m looking forward to working hard this summer and hopefully pushing for more opportunities. We have a fantastic squad and everyone is really excited about what we can achieve in the upcoming seasons.”
Sussex’s head coach Mark Davis said: “It’s really exciting news that both players have extended their contracts. Both have exciting futures and the fact that they are homegrown players is a testament to our youth and academy system. We look forward to seeing how they progress over the next couple of seasons.”