Payet sealed a return to Ligue 1 giants Marseille in late January after stating he would never play for West Ham again.
Following the conclusion of the transfer saga, Lanzini was tasked with the job of trying to fill Payet's boots with his goals, which he has done by scoring three times and creating two others for the Hammers since mid-January.
The Argentine says he isn't feeling the pressure of his new role, however, and insists he is enjoying the added responsibility he now has in the West Ham side.
"Well, I would not say I am feeling the pressure at the moment," the 24-year-old told Sky Sports ahead of the Hammers' home match against champions Leicester City on Saturday.
"Of course, Dimi was a very important player for us and we linked up very well on the pitch. He could help take the pressure off other players with the way he could create space and make things happen.
"But I am very comfortable with the way things are going at the moment and I feel that I am a very important part of the team, which is just what I wanted
"I am feeling fit and confident. The coach has shown faith in me as have my team-mates and that's just what I need in order to show what I am capable of doing on the pitch."