The England international scored the opening goal in Arsenal's 4-0 thrashing of Aston Villa at Wembley on Saturday as the Gunners retained the trophy they won against Hull City the previous season.
In the Premier League, however, Arsenal, who have not won the title since 2004, were never a factor in the title race and finished 12 points behind champions Chelsea in third place.
The Gunners were languishing in sixth place at the turn of the year but finished the season in blistering form and registered impressive away wins against Manchester City and Manchester United in the league and cup respectively.
Those victories fuelled talk that the Gunners had finally added substance to their style and Walcott said the north London side would set their ambitions high next season.
"The Premier League has to be the next target for us," the 26-year-old, who scored a hat-trick in their final league match against West Bromwich Albion last week, told the BBC.
"That's two FA Cup wins now but this is one of the best squads we have had at Arsenal so we should be achieving more. We need to start well in the Premier League next year.
"This team is blessed and there's got to be many more next year."
Walcott's campaign has been disrupted by injury but he has finished the season strongly and he is just one of a plethora of attacking options available to manager Arsene Wenger.
Frenchman Wenger also boasts sparkling attacking players such as Alexis Sanchez, who scored a sublime second at Wembley, Mesut Ozil, Aaron Ramsey and Santi Cazorla.
Combined with an improved steeliness in midfield in the shape of Francis Coquelin and a defence which was only bettered by champions Chelsea and Southampton, talk has increased that the jigsaw maybe slowly falling into place at the Emirates Stadium.
"If we keep this squad, with one or two more players, we will have a good chance to challenge for the title," French forward Olivier Giroud, who scored Arsenal's fourth goal on Saturday, told the BBC.
"The Premier League title is the target. Hopefully we will be there."
(Reporting By Tom Hayward; Editing by John O'Brien)
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