As @TamirEvan stated in the comments, Ben-Sira appears a few times in the Talmud, for example:
"The verse states: “All the days of the poor are terrible” (Proverbs 15:15). The book of ben Sira says: Also the nights are terrible, as then the poor person worries. The poor person’s roof is among the lowest roofs in the city, and in the elevation of the hills is his vineyard, at the highest point, as those are of the lowest quality and consequently the least expensive places for each. From the rain on the roofs of the entire city, water will flow down to his roof and dampen it, and the soil of his vineyard is eroded by the rain and swept down to other vineyards."
and Ketubot 110b:
"The mishna taught: Rabban Shimon ben Gamliel says that a pleasant residence tests the individual. The Gemara asks: What is the meaning of the term tests in this context? The Gemara explains: This is in accordance with the opinion of Shmuel, as Shmuel said: A change in one’s eating habits [veset] or in one’s place of residence is the start of intestinal disease. Similarly, it is written in Sefer Ben Sira: All the days of the poor are terrible. And yet there are Shabbatot and Festivals, when even the poor eat well. Once again, the Gemara answers: This is in accordance with the opinion of Shmuel, as Shmuel said: A change in one’s eating habits or in one’s place of residence is the start of intestinal disease, and as a result the poor suffer even from a change for the better. Since the Gemara quoted from Sefer Ben Sira, it cites the rest of the passage concerning the terrible days of the poor. Ben Sira says: Even the nights of the poor are bad. His roof is at the low point of the roofs, i.e., his residence is at the lowest point in the city, and his vineyard is at the mountain peaks, at the highest point of the slope, which means that the rain of roofs washes down to his roof, and the soil of his vineyard to other vineyards, i.e., the rain washes away the soil in his vineyard and carries it away to the vineyards below."
In both places, it seems that the gemara used Ben-Sira as an "asmachta" (אסמכתא) much in the same way as it usually uses psukim from the Tanach. However, in other places, it says one should not read Ben-Sira.
Rabbi Dr. M. Z. Segal in his edition of Ben Sira, wrote that even teachings such as "Rabbi Levitas a man of Yavneh said: be exceeding humble spirit, for the end of man is the worm." (Avot 4:4) are paraphrases of teachings of Ben Sira, who wrote long before "Abase pride very exceedingly; For the hope of man is the worm" (Ben Sira 7:16) (Compare the Hebrew).
The Talmud and midrash further mention two books that we don't currently have but were included in the apocrypha (Sefarim Chitzonim - External Books, i.e. books that should not be studied) in the time of the gemara: Ben La'anah (Yerushalmi Sanhedrin 50a) and Ben Tigla (Kohelet Rabbah 12:12).
Furthermore, Rabbi Prof. Shnayer Leiman in one of his classes on the Dead Sea Scrolls, mentions that Rashi quoted a verse that doesn't appear in Tanach: "אם תעזבני יום יומים אעזבך - If thou forgettest Me one day, I will forget thee two days". Rashi says it's from a certain "megillah" (scroll). Both the midrash in Sifrei Devarim 48:8 and the Yerushalmi in Brachot 68a say it's from "Megillat Chassidim". It's possible that this scroll was also an apocryphal work in the time of the gemara.
In an essay titled "The Inverted Nuns", Rabbi Leiman quotes Midrash Mishlei 26:24 (emphasis mine):
"The verses beginning: "When the ark was to set out" (Num 10:35) are marked above and below. Rabbi [Judah the Prince] said: It was a separate book and withdrawn."
And then he presents the (reluctant) opinion of Rabbi Shaul Lieberman that this midrash may be referring to a book known as "Sefer Eldad U'Meidad", because there's another midrash that was found in the Cairo Genizah and first published in the late 19th century by Elkan Adler which says (emphasis mine):
"Some Midrashim expound this differently. They state: Why did the sages place
inverted nuns before the verse: The people took to complaining (Num 11:1)? The
sages thereby declared: The entire Torah consists of the prophecy of Moses except
for these two verses (i.e., Num 10:35-36) which are from the prophecy of Eldad
and Medad. Therefore they were enclosed with a curved nun and inserted into the
Torah." (original Hebrew at the bottom of pg. 705)
Rabbi Leiman notes in his essay that a "Book of Eldad and Meidad" appears in lists of apocryphal or pseudepigraphal works but is currently non-extant, although per these sources, it seems that this work may have been incomparable to other External Books, because it was actually held, at least by some authorities, to have been written by Eldad and Meidad (on what this book may have contained, see Sanhedrin 17a and Sifrei Bamidbar 95) (it should be noted that Rabbi Leiman himself believes that both this formerly-lost midrash and another midrash call Midrash Chaserot V'yeterot both misinterpreted Midrash Mishlei, which he believes does not refer to a lost book authored by Eldad and Meidad) (however, Prof. Baruch A. Levine wrote an essay attached to the end of the PDF of Rabbi Leiman's essay, in which he argues against Rabbi Leiman's conclusions).
In Jahrbuch IX, pg. 13-15, Rabbi Dr. Heinrich Ehrentreu wrote that he thinks the verse from Yevamot 86b, brought by @moses in his answer, "ושוטרים הרבים בראשיכם", is from an apocryphal work, specifically, in his opinion, the original Hebrew version of Maccabees I, which he believes is the same work as the "Megillat Beit Chashmonai" mentioned in the Halachot Gedolot as having been written by sages from Beit Hillel and Beit Shammai. There are a number of other interesting interpretations of the verse. Rabbi Amos Sofer in Sinai, vol. 98, thought it might be from one of the Dead Sea cults or perhaps the Essenes (which may or may not be the same group).
Other than this verse, there are multiple verses in Chazalic works that don't appear in Tanach. I've managed to find 23, but there are likely others. Most commentators think that they are either shortened version of extant Tanachic verses or come from Ben Sira, but a few - mostly modern scholars - think that at least some come from apocryphal works.
There are also parallels that can be drawn between certain apocryphal works and midrashim. For example, Professor Yehoshua Grintz (in his edition of Sefer Yehudit), Chanoch Albeck (Beresheet Rabbti, pg. 15) and Adolf Neubauer (Book of Tobit, pg. 38) all write that the midrash in Tanchuma Ha'azinu 8 is clearly the story of the Book of Tuviyah with the names and much of the plot removed, while Rabbi Dr. Moshe Gaster found an edition of Megillat Ta'anit which includes the story of Yehudit. The difference between this version of the story of Yehudit and other Chazalic mentions of the story is that this version adds a "Yom Tov" that they celebrated which is similar to the celebration mentioned at the end of the Aramaic version used by Jerome in the Vulgate (this "Yom Tov" isn't mentioned anywhere else) (see here for more information).