Tavernier laid on a brilliantly-worked equaliser for Paddy McNair on the hour before he scored the winner himself nine minutes later to seal all three points and prevent Steve Bruce from enjoying his first success as West Brom manager.
Jayson Molumby, on loan from Brighton, had given the Baggies the lead in the 28th minute at the Riverside as Albion found the target for the first time in six matches.
However, ending the 479-minute goal drought was not enough for Bruce to celebrate a victory in his fourth match in charge, leaving West Brom 11th and six points off the play-offs.
Boro, who have won seven games in a row at home, climbed back into the top six courtesy of the turnaround, which followed a frustrating start for the Teessiders.
When referee Michael Salisbury deemed there was not even a foul by West Brom full-back Conor Townsend, who appeared to pull down Isaiah Jones early on, the home side’s frustration grew.
Boro’s best opportunity in the opening period came when goalkeeper Sam Johnstone spilled an effort from Riley McGree.
After that the hosts struggled to get going and the players and fans questioned other decisions from Salisbury, who was booed off with his assistants at the break.
There was also some hesitancy and sloppy play at the back from the home defence and goalkeeper Joe Lumley which disappointed the crowd and manager Chris Wilder.
West Brom exploited that on one occasion midway through the opening period when Andy Carroll managed to roll Dael Fry to get to the byline.
Carroll’s low ball into the six-yard box was half cleared as far as Molumby, who struck first time into the bottom corner of the net for his first goal for the club. In the second half, Wilder introduced Duncan Watmore and Andraz Sporar to his forward line and 15 minutes later the turnaround was complete.
Middlesbrough: Lumley, Dijksteel, Fry, McNair, Jones, McGree, Howson, Tavernier, Taylor (Watmore 55), Balogun (Sporar 55), Connolly (Peltier 83). Unused subs: Bamba, Lea Siliki, Daniels, Coburn.
West Brom: Johnstone, O’Shea, Ajayi, Bartley, Townsend (Furlong 82), Mowatt, Molumby, Diangana (Tulloch 80), Reach (Robinson 77), Ahearne-Grant, Carroll. Unused subs: Clarke,Kipre,Button,Gardner-Hickman.
Referee: M Salisbury (Lancashire).