Verb + 着 (zháo) (managed to / succeeded)

(ongoing -ing)着zhe (zháo) — "managed to / got it"

• Verb + (ongoing -ing)着zhe read zháo marks success: the action hit its mark.
• to fall asleep睡着shuì zháo(completed / change)了le = "fell asleep"; to try to find找zhǎo(ongoing -ing)着zhe(completed / change)了le = "found it"; to buy买mǎi(ongoing -ing)着zhe(completed / change)了le = "managed to buy it".
• Don't confuse it with (ongoing -ing)着zhe (zhe), the ongoing-state marker.

Verb + (ongoing -ing)着zhe (zháo)

• (ongoing -ing)着zhe (zháo) says the verb succeeded in reaching its result.

he他tāfinally终于zhōng yúsleep睡shuì(succeed)着zháo了le。

• to fall asleep睡着shuì zháo(completed / change)了le = "(he) fell asleep."

Recap

Read the full reference: HSK 5 grammar — Complements.