Anthony's contract also includes a rare no-trade clause and has an early termination option after the fourth year, according to league sources.
Anthony was eligible to sign a contract for a maximum of $129 million but agreed to a deal worth $5 million less to give the Knicks more financial flexibility to pursue free agents in future summers.
Anthony will make a maximum salary of $22.458 million in the first year of his contract, which increases gradually each year to a peak of more than $27.9 million annually.
Sources told ESPN's Marc Stein that Anthony will make $22,875,000 in the second year of his deal, $24,559,380 in the third, $26,243,760 in the fourth and $27,928,140 in the fifth and final year.
The raise of roughly $400,000 between the first and second years of Anthony's contract is $1.3 million less than the maximum raise he was eligible for.
This is pivotal for the Knicks because they hope to attract big-name free agents in the summer of 2015. Anthony's taking less than the maximum in the second year of his contract will give them extra money to do so.
Anthony also has an early termination option after the fourth year of his contract, sources say. That will allow Anthony to test free agency in the summer of 2018 if he so chooses.
In addition, league sources confirmed Anthony has a no-trade clause in his contract, which is rare in the NBA. The clause was reported by the New York Daily News earlier Tuesday.
Anthony first mentioned in February the possibility that he would accept a contract worth less than the maximum in free agency.
"Without a doubt," Anthony said while in New Orleans for All-Star Weekend. "Any opportunity I have to build that up in New York, I'd do it. I told people all the time, always say, 'If it takes me taking a pay cut, I'll be the first one on [Knicks owner] Mr. [James] Dolan's steps saying take my money and let's build something strong over here.'"
Knicks president Phil Jackson publicly challenged Anthony on that point in recent weeks, saying several times that he hoped Anthony would accept less than a maximum contract to give the Knicks financial flexibility.
"I think [there is] a precedent that's been set," Jackson said shortly after the end of the regular season. "Because the way things have been structured now financially for teams is that it's really hard to have one or two top stars or max players, and to put together a team with enough talent, you've got to have people making sacrifices financially.
"So we hope that Carmelo is true to his word, and we understand what it's going to take, and we will present that to him at that time."
Jackson ended up offering Anthony a maximum contract of $129 million over five years when the Knicks met with him during his free-agency visits. The offer was one of five contract options presented to Anthony, according to the Knicks president.
Jackson insinuated Sunday that Anthony accepted a deal worth less than maximum money in the early seasons to help the Knicks' financial flexibility. Anthony announced he was re-signing with the Knicks on Sunday afternoon.
"He did exactly what we kind of asked him to do," Jackson said. "Give us a break in the early part of his contract so that when we have some wiggle room next year, which is hopefully big enough wiggle room, we can exploit it."
Portland's LaMarcus Aldridge is eligible to test free agency next summer but has said that he intends to re-sign with the Trail Blazers. Minnesota star Kevin Love also could test free agency next summer but he is expected to be traded by the Timberwolves to a team with which he is comfortable signing a long-term contract.
Information from ESPN.com's Marc Stein was used in this report.