The goals secured three points that meant the Dons can no longer catch the Ibrox side in second place and keeps Steven Gerrard's side's slight hopes of catching Celtic alive.
Tavernier's double saw him reach 16 goals for the season with 14 arriving from the penalty spot.
The award of two penalties by Don Robertson for challenges on Nikola Katic brought the number of penalties won by Rangers this season to 17.
While Celtic's all arrived in domestic competition, Rangers were awarded two in Europa League qualifying and one in the Europa League group stages.
Of the 17 penalties, 16 have been taken by Tavernier. Fourteen were scored while the two which were missed arrived in the space of seven days, in the 4-0 win over St Mirren in the league and 0-0 draw with Kilmarnock in the Scottish Cup.
Jermain Defoe took and netted the other spot kick.