The 20-year-old moves to Welford Road from the New South Wales Waratahs having previously played for Parramatta and West Harbour.
Mafi's time in Australia saw him represent Australia Schools and Under 20s before he made his senior international debut for Tonga against Samoa in June this year.
"We've got good competition right through the squad but injuries to Richard Blaze and Geoff Parling have left us a bit thin in the second row. It is very close to the start of the season and I'm delighted that we've been able to bring in Steve," said Tigers boss Richard Cockerill.
"He has come through the ranks within Australian club rugby and has been very highly regarded there.
"He's a big, strong guy, he's young and ambitious, and we're looking forward to seeing how he settles into his rugby here."
But while Mafi could feature in Leicester's opening Heineken Cup clash with at Benetton Treviso on Saturday, October 9, Crane is unlikely to be involved in European action until the Round 3 encounter at Perpignan on December 11.
The 24-year-old, who kicked the winning penalty in the 2009 semi-final shootout against Cardiff Blues, suffered an ankle injury in a pre-season game in Munster and has since had surgery to correct the problem.
"Jordan has had surgery on some ligament damage in his ankle to clean that up," Cockerill told the Leicester Mercury.
"It's a bit worse than we first feared."