Verb for 'luck'

Maybe you could use "lucked out" meaning

to be very lucky

[Cambridge English Dictionary]

It's considered a "phrasal verb with luck verb".

Might work for you.

Actually, "I lucked through my exams" is exactly how you would say it, at least according to The Free Dictionary, Vocabulary.com and my own personal experience.

"I really won the lottery when I got through those exams." [1]

[1] Note: in this case, "really" is just used for emphasis, and not actually meaning that a person really did win a lottery.

"I pulled off a miracle when I got through those exams."

"I was fortunate through those exams."

If you were just a lucky guesser, "I guessed my way through those exams."

There is also the concept of verbing a noun. (Interestingly, the word "verb" refers to a type of word, which is a noun, so the word "verb" is a noun, so the phrase "verbing a noun" is an example of verbing a noun.) If you can think of any familiar examples, whether historical and global or even recent and local, of a lucky person or event, you could use a word that refers to a lucky person or event. For example, "I Stevened my way through those exams last week" might not be readily understood in general, but if the conversation was just discussing how Steven spectacularly pulled off a streak of luck in four different ways over the past few days, then associating Steven's name with the word "luck" could be understood to anyone familiar with the context.

I got lucky with my exams (I see this was also noted by @mjf in comments above). (From Merriam-Webster, "to have good luck : to succeed because of good luck" (note the close association with sex, as noted in previous comments: "get lucky" also means "to succeed in finding or getting someone to agree to have sex with one" — although I don't think there's any danger of confusion/misinterpretation in this context).

(As a phrasal verb, this answer arguably this does not satisfy the [single-word-request] tag, but the original example, "lucked through", is also a phrasal verb ... as are many of the other answers here ... "lucked up", "lucked out", not to mention "won the lottery". Of the current answers, only "fluked" [which is not common in my experience in AmE!] is a single word ...)