Chambers had been strongly linked with a switch to the Emirates Stadium in recent days, and Southampton confirmed on Monday that the full-back has moved to north London for an undisclosed fee.
Arsenal subsequently revealed the 19-year-old - a graduate of Southampton's famous youth academy - has signed a long-term contract with the club.
In a statement released on Arsenal's official website, Chambers said: "I am so happy to be signing for Arsenal. They are a team I have much admired for their playing style, and a team who have been one of the top sides in Europe for many years.
"I'm looking forward to joining up with my new team-mates today and beginning preparation for the season ahead.
"I would like to take this opportunity to thank everyone at Southampton FC for everything they have done for me and for making this move possible."
Arsenal boss Arsene Wenger also expressed his delight at securing Chambers' signature.
"We are very pleased that Calum has agreed to join us. The way he adapted to the Premier League last season with Southampton at a young age shows that he has tremendous quality," said the Frenchman.
"He has a lot of the attributes that we look for in a young player and I am sure that he will do well with us."
Chambers - who enjoyed a breakthrough season in the Premier League in 2013-14, making 22 top-flight appearances - is the latest in a long line of departures from Southampton since the end of last term.
Captain Adam Lallana, centre-back Dejan Lovren and striker Rickie Lambert have all signed for Liverpool, while left-back Luke Shaw has gone to Manchester United.
Manager Mauricio Pochettino has also been prised away, the Argentinian having taken over from Tim Sherwood at Tottenham, while Morgan Schneiderlin and Jay Rodriguez have been linked with moves away from St Mary's Stadium.