And the manager also praised his staff at the training ground after he received the Manager of the Month award for the same period, insisting the team's recent impressive form and results have been a group effort.
Tavernier scored four goals in the league in November plus one more in the 4-0 League Cup win over Falkirk. His tally now sits at 15 goals for the season, an incredible figure for a right-back.
Gerrard said: "He's shown fantastic leadership on and off the pitch at the moment James. I think he's really grown into the role.
"He's wanting to improve as a player on a daily basis which is the most important thing, but he's also aware that he wants to grow and learn more as a leader as well.
"In terms of our award as a staff, it should really be called a management team award because everyone puts so much in, not just the management staff but behind the scenes there are a lot of staff who push the team and do everything they can to take all the excuses away from the team.
"But we wouldn't get this award without the consistency of the players, they deserve the plaudits and it's an award that is certainly shared around the club really."
YOU CAN WATCH THE FULL PRESS CONFERENCE WITH JAMES TAVERNIER HERE
One player Gerrard was keen to talk up in this morning's pre-match press conference was Kemar Roofe.
The Englishman put in a great shift for his side last night as the Light Blues beat Standard Liege to progress to the Round of 32 in the Europa League, and his tireless work didn't go unnoticed, with his gaffer saying: "One thing is for sure, I have to pay credit to the fitness staff and the sports science lads.
"The numbers that we are hitting physically at the moment are really impressive and I think Kemar Roofe certainly helps on that.
"He doesn't stop. He's like a Duracell bunny and he's hitting numbers that you don't really expect from a forward player. He's running more than the midfielders and he's running the most in the team at the moment.
"I think he's really setting the example in terms of what we need to put into a game to get the results we are getting. Everyone is hitting impressive numbers, but Roofe has taken and raised the bar the last couple of performances he has put in."
Rangers are now preparing for Sunday's Premiership match against Ross County in Dingwall and Gerrard provided an injury update ahead of the trip to the Highlands.
He added: "We'll give Ross County respect and we're ready for a tough challenge, but what I would say is the focus is always on Rangers and we couldn't be in a better place going into this game.
"We've got one or two missing but in the main we're going up there with a healthy group. Jack has not progressed as quickly as we would like, so I don't think he'll be available for the weekend and Helander is still missing for obvious reasons.
"Everyone else should pull through from last night. We have got minor knocks but that shouldn't rule them out for the weekend."
YOU CAN WATCH THE FULL PRESS CONFERENCE WITH STEVEN GERRARD HERE