Seven Maori players were making their debuts, with four more newcomers in the reserves, but the Tigers were missing 10 internationals.
Maori halfback Frae Wilson scored his side's first try in front of a crowd of 17,000 but the visitors were quickly on the back foot as they conceded tries to Tigers wings Niall Morris and Adam Thompstone.
At 26-7 down, the Maori were looking at a demoralising defeat but a Willie Ripia penalty and converted try from second-five Tim Bateman put them back in touch at 26-14. Another Ripia penalty narrowed the halftime gap to 26-17.
Centre Charlie Ngatai scored the only try of the second half but the Tigers' defence and regular errors from the Maori conspired against Joseph's men.
Their next match is against a Championship Select XV on Sunday (NZT), before they play Canada at Oxford University next Saturday.
Leicester Tigers 32 (Niall Morris, Adam Thompstone tries; George Ford 2 cons, 6 pens) NZ Maori 24 (Frae Wilson, Tim Bateman, Charlie Ngatai tries; Willie Ripia 3 cons, pen). HT: 26-17.