Final answer:
In the context provided, 'got round' means to 'get your way with' someone by persuading or influencing them.
Step-by-step explanation:
In the passage from Liza, the phrase "got round" her means to 'get your way with' someone, usually by persuasion or manipulation. Mrs. Pearce is indicating that despite wanting to leave the speaker, she has always been convinced to stay at the last minute because the speaker knew how to persuade her effectively. This phrase implies a level of cunning and ability to influence someone to act against their prior intentions.