Up until the closure of the transfer window, Sanchez - who has less than a year remaining on his contract - was strongly linked with a move to either Manchester City or Paris Saint-Germain.
However, Sanchez returned to the starting lineup for Monday's 2-0 win over West Bromwich Albion in the Premier League, and Koscielny has claimed that the Chilean will only continue to improve over the coming weeks.
"Alexis is an important player for us. I know it was a difficult situation with the transfer window but now it's finished and I think he is focused on his job, on his performance and on his club," The Express quotes the 32-year-old as saying.
"He's an Arsenal player and he knows that he needs to play and he needs to score some goals and give assists to be better.
"He's still working every day like he did before, so the most important thing is to keep his head cool and just play and enjoy."
Sanchez has netted just one goal in six appearances in all competitions this season.