I felt undermined at work today

Not sure if this is the right place but let me share just in case.

So, I just got promoted into a really high managerial position this week and already I have had my first disgruntled member to deal with. I asked my direct subordinate to ask another employee (call him Mr.Delivery) to do a delivery run(as is his job duty) but he cited that he was busy. He was also busy for another reason yesterday when we reached out to him.

MY manager(call him Mr.Manager) happened to overhear this conversation and stepped in and talked to Mr. Delivery's manager(Mrs. Sub-Manager) asking if he was busy to which the manager stated that he was free. Hence, we asked Mrs. Sub-Manager to send Mr. Delivery for our urgent work.

So, this employee shows up at my office demanding to see me. A bit of raised voices and saying that I complained to his manager that he was lying about work and he is actually busy and this made him look bad. It appears that Mrs. Sub-Manager scolded him about lying about work before sending him. He agreed to do the work but before he left, he said something along the lines of "make sure this doesn't happen again".

He did this all in front of MY direct subordinates and my new staff who joined recently. I was so stunned and felt like a deer caught in headlights that I didn't yell at him or anything. I was quite overwhelmed.

I also felt that my authority was very undermined - that a much lower ranked subordinate(and I am not trying to put him down here but I just want to explain the hierarchy here) spoke to me in such a way.

MY manager caught wind of this and told me I shouldn't have let Mr. Delivery speak to me like that since I am way above his station and I should have put him in place. I am new to being a manager so this felt so oddly devastating like I was totally out of control. That it happened perhaps because I am young and female too. Or just simply because I am a pushover and not standing up for myself.

I wish I had been more firm with him instead trying to cower and placate him.

I don't know if I want advice here but if anyone has any, thats good too. But I have been feeling really odd since this incident.