The Tigers played a full part in Saturday's Wembley spectacular, taking a 2-0 lead over Arsenal before being pegged back and eventually going down 3-2 in extra-time.
Europa League football beckons for the club next season and Rosenior thinks they have to look to grow and push on to secure more chances of silverware in the near future.
They impressed in spells during their first season back in the Premier League as well, with a 16th-placed finish enough to secure their main aim - survival.
"We want to be successful and we can't rest on our laurels now," said Rosenior. "We want to grow as a club, we don't just want to stand still.
"The Premier League is the best league in the world and we want to make sure we are stabilised in it for years to come. We also want to come back here and have more days like this one.
"I'm proud to play for this football club because it's a special one and the fans are fantastic.
"Hopefully we've done them proud. When the time comes I'm sure I'll enjoy this too but at the moment it still hurts."
Midfielder Stephen Quinn admitted his pride at their positive season and also paid tribute to the club's fans, who have been through a lot this season over owner Assem Allam's move to change their name.
"We're so proud of each other and I'm proud of all the lads in that dressing room," said Quinn. "We were so close but when you have top quality opposition like Arsenal there's no shame in losing.
"The fans were excellent and the last couple of trips to Wembley have been great for them. They made a big sound and I'm sure they had a good day.
"They know how hard we've worked and I'm sure they're proud of all the boys. Getting safe in the Premier League and reaching the FA Cup final, it's the best season in their history."