The France international will join the Blues on an 18-month contract with an option to extend it for another year after the two clubs agreed a deal on Wednesday.
The move has now been made official after Arsenal confirmed the signing of Pierre-Emerick Aubameyang from Borussia Dortmund, who have just loaned Michy Batshuayi from Chelsea.
A triangle of transfers developed between the three clubs over the last few days of the January transfer window, with each move seemingly hindering on one of the others going through.
While Dortmund wanted Batshuayi as Aubameyang's replacement, Chelsea were reluctant to let him go without bringing in another alternative to Alvaro Morata and turned to Giroud after failing to land Edin Dzeko from Roma.
Arsenal, however, were awaiting their club record signing from the German side before relinquishing the France international.
Giroud told Chelsea's official website: ‘Chelsea is one of the biggest teams in the Premier League. They have won maybe the most trophies in the last six or seven years. It’s a massive club and I am proud to sign for Chelsea, and I’m looking forward to playing.’
The 31-year-old had grown frustrated at the Gunners after being dropped to the bench following Alexandre Lacazette's arrival from Lyon.
Giroud started just one of his 16 Premier League games this season, scoring four goals, and had been linked to West Ham, Everton and Borussia Dortmund in recent months but opted to stay at the Emirates Stadium.
Although first-team opportunities were rare, Arsene Wenger urged him to stay at the club for the remainder of the season.
But the forward admitted in December that a move in January was likely, with a spot in the France squad at the 2018 World Cup in his sights.
“I have the World Cup at the back of my mind weekly, almost daily,” Giroud said last month.
“Each time I go onto the pitch, it's to be more clinical and keep on track for the World Cup.
“If I wasn't able to leave this summer, it was because of the opportunities I had... but it's certain that eventually, even in January, I'm going to have to ask myself the question again."
Wenger has spoken of his respect for Giroud, telling the club's website: “Oli is a guy who gave great service to the club, he never let us down on the commitment front and he saved us many times,” said Arsene Wenger.
“We signed him from Montpellier and he became a strong, regular international player for France, and we have gone together through good and bad. But always I believe that his behaviour has been top, top class.
“He always had his heart 100 per cent in the team and he was committed.”
Giroud departs north London with Arsenal sitting sixth in the Premier League, while Chelsea sit third, three points behind Manchester United.
Giroud arrived at Arsenal in 2012 as a Ligue 1 champion with Montpellier and went on to make over 250 appearances for the north London side, scoring 105 goals and winning three FA Cups with Arsene Wenger's side.