As a non-league side Boro famously held Premier League Newcastle to a 1-1 draw in 1999 and also beat the Magpies three seasons ago.
In 2012 they reached the fifth round for the first time and took Tottenham to a replay before going down 3-1 at White Hart Lane.
On Saturday they take on 2008 winners Portsmouth, who are now a division below, and Westley has his sights set on another cup run.
"Everyone loves the FA Cup, it's special," said Westley.
"It's been central to this club's evolution in recent years, we've had some big runs and valuable runs.
"It's up to us to win the match and get through to the second round."