Mourinho questioned Lampard's decision to leave a number of experienced players out of his starting line-up at Old Trafford, including the half-fit N'Golo Kante, with Chelsea collapsing to a 4-0 loss.
The defeat marked an unwelcome start to Premier League management for Lampard, but Chelsea's all-time leading goalscorer does not plan on getting involved in a slanging match with the Portuguese, who coached him at Stamford Bridge.
"I was very ready for that," Lampard said ahead of Monday's (AEST) game against Leicester City.
"Being a player who played here, we were always competitive for leagues and [UEFA] Champions Leagues. The level of attention and criticism on your performance is inflated as a manager because you take on more responsibility.
"I wasn't surprised by it. You have to be ready for it. You have to be ready to answer questions. You have to be ready to know what your feeling is. Whether you should say something bad.
"My first feeling is can I protect my players? Can I protect my club? Can I protect myself? Because I know we are working with the right intention and in the right direction. I know you have to pick and choose your battles."
Chelsea also slipped to defeat in midweek as it was edged out on penalties by Liverpool in the UEFA Super Cup.
Christian Pulisic, Olivier Giroud and Kante came into the side as the Blues delivered a much-improved performance in a game that finished 2-2 after extra-time.
Lampard's team had a more solid look about it with Kante in midfield but the 41-year-old wants his tactics to be adaptable.
"I think it will evolve," Lampard said of his game plan. "I hope so because you need to be thinking to evolve all the time. I don't want us to be a one-trick pony. I want us to have different strengths to our game.
"Liverpool, for instance, have one of the best front threes in the world [Sadio Mane, Roberto Firmino and Mohamed Salah]. We know that. A lot of speed, a lot of quality to hurt you and so we have to think about that and change the approach to do that. I don't think there is one certain shift.
"What I did love was the way the players took on the information we gave them. There was a really short space [of time] from Manchester United. Then there were a few messages, and they took it on brilliantly.
"There were generals on the pitch. I thought Jorginho was fantastic the other night. He was constantly talking to people around him. Constantly talking to the back-line to drag them up and playing with quality. They are the players who are the ones that drive when they get out there.
"That was the most pleasing thing from me - that the spirit and attitude of the players felt really special."