Arsenal ended a nine-year trophy drought with the FA Cup last month but although they led the Premier League for the majority of the season, they were heavily beaten by several of their title rivals and eventually finished fourth.
The departure of right-back Bacary Sagna has created a void in that position, and the mid-season injury to Aaron Ramsey exposed a lack of depth in midfield.
But it is in attack where Arsenal are the lightest, with Olivier Giroud their only senior striker.
"We'll need a goalkeeper, a right back, a midfielder and a striker," Koscielny said at France's pre-World Cup training camp.
"It's important to have a minimum of these four players so that we'll be better able to challenge over the course of a whole season against teams like Manchester City, Chelsea or (Manchester) United. Winning the FA Cup can change things and lead to other players coming."
Giroud scored 16 league goals but Arsenal's over-reliance on him is evidenced by the fact he started 36 of the club's 38 league games.
The Gunners have been linked with a move for QPR striker Loic Remy, who spent last season on loan at Newcastle United, and AC Milan's former Manchester City maverick Mario Balotelli.
"We need to recruit a very, very good striker, because it's important for us, for competition for places, because Olivier needs that to become even better," said Koscielny, who recently signed a contract extension until 2019.
"Having a striker who scores more than 30 goals a season can help us win the Premier League. Compared to the squads like Manchester City, even United, Chelsea, you can see there's a certain difference."
Koscielny likened the FA Cup victory to a great weight being lifted off the team's shoulders and feels the fact they have ended their trophy drought will make them better next season and beyond.
Arsenal beat Hull 3-2 in extra time in a thrilling FA Cup final. Defeat would have been hugely damaging for Wenger, who was under severe pressure to deliver.
"It was a relief for everyone. We felt a weight coming off us. We made the most of it when we celebrated our trophy, it was an extraordinary moment," he said.
"Thanks to this trophy, we'll be able to play more freely in the years to come without this pressure hanging over us, thinking 'we must win a trophy.'"