The right-back prevailed from an all-Ibrox shortlist following the club's Scottish Premiership title win.
Tavernier, who scored 19 goals this season, beat team-mates Allan McGregor, Ryan Kent and Steven Davis to the prize.
Davis earlier scooped the Scottish Football Writers' Association award while McGregor was named SPFL Premiership player of the year earlier in the week.
Steven Gerrard clinched the GMB Scotland manager of the year award, selected by his peers, to make it a treble of all the major manager prizes.
Gerrard succeeded from a shortlist that also contained St Johnstone's Callum Davidson, who is one game away from a cup double, Livingston boss David Martindale, and John McGlynn of Raith Rovers.
Celtic midfielder David Turnbull won the PFA Scotland young player of the year prize.
The former Motherwell player beat Rangers' Nathan Patterson, Ali McCann of St Johnstone and Hibernian left-back Josh Doig to the award.
Hearts striker Liam Boyce won the Championship player of the year while the Scotland national team won a special merit award for lifting the country's spirits by reaching the Euro 2020 finals.