Of course “Lucky !” is English word, but Japanese people say “Lucky !”/「ラッキー!」very often when a good thing happens unexpectedly. This is a casual expression. If you want to express it politely, you should say “Lucky desu”/「ラッキーです」about your current things, “Lucky deshita”/「ラッキーででした」about your things in the past, “Lucky desune”/「ラッキーですね」about other people’s current things or “Lucky deshitane”/「ラッキーでしたね」 about other people’s things in the past. There are other expressions Japanese people use in daily conversation. I will introduce them.
You can say it in a casual situation. If you want to express it politely, you should say “Tsuitemasu”/「ついてます」about your current things or “Tsuitemashita”/「ついてました」about your things in the past. When you say about other people’s things, commonly you put “Ne”「ね」to the last of the sentence. I will introduce the examples.
A:It is unusually sunny today although it was rainy the other days this week. You can enjoy watching baseball game./今週、雨だったのに今日は珍しく晴れだね。野球観戦を楽しめるね。
B:Yes ! I’m lucky !/うん！ついてる！
A:I went to buy sushi yesterday and got a larger serving of it for free !/昨日、寿司を買いに行ったら、無料で大盛りにしてくれたよ！
B:Wow, you were lucky !/おー、ついてましたね！
I would say “Tsuiteru”/「ついてる」has other meanings such as with, on or attached. As you may be confused, I will introduce the example because you need to know that to use “Tsuiteru”/「ついてる」well.
A:There is the rice on your face./顔に米がついているよ。
B:I didn’t notice that. Thank you./気づかなかった。ありがとう。
This is a casual expression. You can say politely “Ungaiidesu”/「運がいいです」about your current thing or “Ungayokattadesu”/「運が良かったです」about your things in the past similarly to “Tsuiteru”/「ついてる」. If you put “Ne”/「ね」to the last of the sentence, you can say about other people’s things. I will introduce the examples.
A:You picked up 500 EURO on the street last week, right ?/先週、道で500ユーロを拾ったんだよね？
B:Yes, I was lucky./うん、運が良かった。
A:My neighbour moved to somewhere and I got his piano today./隣人が引っ越して、今日ピアノをもらったよ。
B:Wow, you are lucky!/おー、運がいいね！
Japanese people say often “Lucky”/「ラッキー」, “Tsuiteru”/「ついてる」 or “Ungaii”/「運がいい」which mean “Lucky” in daily conversation. All of them have the same meanings and nuances. “Tsuiteru”/「ついてる」also has the meanings of with, on or attached.