The veteran wide receiver signed a deal to play with star quarterback Andrew Luck in Indianapolis, the team confirmed Wednesday.
Johnson also tweeted about his new home:
A source told ESPN's Josina Anderson that the deal is for three years and $21 million.
"I felt like this was the best place where I can win a championship and help win a championship," Johnson said. "That's what it's all about. That's why we play the game."
Johnson was released by the Houston Texans earlier this week, ending his 12-year tenure there.
The Texans previously had given Johnson permission to seek a trade, while the seven-time Pro Bowler asked the team to release him.
After 12 seasons with Houston -- all of them as a starter and many as one of the best receivers in the NFL -- Johnson's role on the team was set to change. Rather than accept that, he requested to be traded or released.
"I knew it for a while," Johnson said of his departure. "Knew it even before the season started."
Johnson had two years remaining on his contract with the Texans and a salary-cap figure of $16.1 million for 2015.
"It bothered me a little bit because you have people who tell you they want you be a Texan for life and at the end of year, they let you go," he said.
Johnson's production dipped in 2014 as second-year receiver DeAndre Hopkins emerged. Hopkins notched his first 1,000-yard NFL season, while Johnson caught only 85 passes for 936 yards. In each of the previous two seasons, Johnson had caught more than 100 passes, for 1,407 and 1,598 yards.
Johnson had been the longest-tenured player for the Texans, having been drafted third overall in 2003, the expansion franchise's second draft. In his time with the Texans, he caught 1,012 passes on 1,599 targets for 13,597 yards.
The Colts have added five players to their roster since free agency started Tuesday.
The team signed veteran running back Frank Gore to a three-year, $12 million contract. Johnson and Gore both played at the University of Miami.
ESPN.com Titans reporter Tania Ganguli and ESPN.com Colts reporter Mike Wells contributed to this report.