This year is 5772 / תשעב (Taf-Shin-Ayin-Bet) on the Hebrew Calendar. What is the earliest reference to the Year counting from the Creation of the world?
Seder Olam Rabbah, by R. Yosei ben Chalafta (2nd century), gives a unified chronology from Creation until his own times (although the last part of it, covering the Second Temple era and its aftermath, is given pretty short shrift).
The Gemara (Avodah Zarah 9b) quotes a baraisa (also from, at the latest, the 2nd or early 3rd century) that makes a prediction for "the year 4231 after the creation of the world."
So evidently this counting was familiar by that period (which corresponds to the 3800s or 3900s since Creation), although it didn't become widespread until later.
(There is also a source - will have to look it up - that the kohen gadol, after having performed the Yom Kippur service and survived going into the Kodesh Hakodashim, would commission a commemorative plaque giving his name and the year since Creation. That would push the use of this era much earlier.)
There was an article published recently in the periodical HaMa'ayan (Teves 5773 pp. 5-19) about this subject. The author quotes sources from the gemara and ge'onim in which they count years this way. However, he concludes that this wasn't the common way to count years at the time. Azaryah Min HaAdumim was the first the cast doubts on the age of this dating system; he writes that it probably originated either sometime after Rabbi Hilel HaNasi (4119/670) or after Rav Sherira Ga'on (~4350/900).
The answer: we started counting Hebrew years in תש"ח - 1948. My answer is completely unsourced, but based on my little research on Masechet Rosh Hashanah.
The bottom line is that the State of Israel is the first time in the history of the Jewish people, when the years from Creation are publicly and officially COUNTED (as opposed to CALCULATED). According to Israeli law you can use Hebrew dates in all formal interactions with any institutions in Israel, including courts, banks, government etc.
In other words, we do have a long tradition of calculating Anno Mundi dates and previous answers brought the sources. However, in my humble understanding, NEVER in the history of Judaism this calendar was publicly and unanimously accepted. Sages throughout the ages kept different traditions of calculating the timeline, but never really "COUNTED". Maybe, because there's no Mitzvah of announcing the year, for example, on Rosh Hashana. There's no obligation to use then in contracts between Jews, so those years in dates served only as "secondary" to secular counting.
Hope it helps a bit.