The 23-year-old, who was picked up by the reigning champions for $866,000 in February's auction, cut and dislocated the little finger on his right hand in the Black Caps' World Twenty20 defeat to Sri Lanka last week.
"There's no fracture, no break and we're very happy with how the injury is going," a relieved Anderson told reporters.
"The ligaments are all fine too; everything else is good to go. Everything went well and the news was all positive."
Mumbai begin the defence of their IPL title against the Kolkata Knight Riders on Wednesday April 16.