Just to have an answer instead of just an edit/addition to the question:

EDIT: PROBLEM RESOLVED:

My Arduino is connected to my PC via USB, so I changed pins for communication with the ESP by SoftwareSerial to pins 10 and 11. Now it works. Thank you.

Ok... sure... your Uno's Atmega384 was talking with pins 0 and 1 to the USB adaper chip (Atmega16u2 on Uno R3) and at the same time to the ESP-01.

Read the fine schematic: Arduino Uno R3