For the first time in 13 months oil is flowing to the rest of the India from the troubled northeast state of Assam, ending a $1.375 billion drain on the nation's already depleted foreign reserves and signaling a possible solution to unrest that at one time carried the threat to secession.

While it is still too early to say whether Prime Minister Indira Gandhi's government has succeeded in overcoming one of its thorniest domestic problems, the end of the blockade that cut the shipment of oil from Assam is the most visible sign that the uprising has lost momentum.

The Assamese students who sparked the popular movement to preserve their state's ethnic identity against large-scale "foreign" immigration have offered to meet with the Gandhi government in what the Hindustan Times newspaper termed "a major shift" from their previous stance.

"They are tired; they want to find some way out," said a native of Assam who lives here but keeps in close contact with the student leaders.

The problem now for the Gandhi government, according to diplomatic observers, is to move quickly for a solution that leaves the Assamese with some measure of self-respect.

"If she handles it right," a diplomat said, "Mrs. Gandhi can emerge from this looking very statesmanlike. If not, it could boomerang into a worse situation with local instability and a low level of violence."

It is especially difficult to gauge what is happening in Assam, an isolated state connected to the rest of India by a narrow neck of land, because foreign correspondents are not allowed to go there.

Furthermore, the new chief minister installed by Gandhi, Anwara Taimur, has imposed a form of press censorship on papers and magazines in Assam, which makes it harder for people outside the state to judge what is going on there.

Nonetheless, reports from correspondents of Indian national newspapers who have visited the state point to an end to the mass uprising that last summer appeared to have the almost total support of the Assamese.

"Oil is flowing once again, plywood and jute can move out of the state without hindrance and schools and colleges are open," wrote Gautam Adhikari from Assam's capital of Gauhati in the Madras daily, The Hindu.

"Trade union activities have restarted and political parties, in hibernation for one year, have resumed functioning in the open. Children and adolescents no longer troop off in hordes for parades and pickets. . . . The people of the state seem to be tired of disruption in normal life."

The peace of Assam was rocked last week by four bomb blasts, one of which blew a hole in the oil pipeline and stopped the flow of crude for 12 hours. But these blasts have been blamed on an extremist fringe, perhaps acting out of frustration because their movement appears stalled.

As a result of the bombings, security has been tightened throughout the state, especially along the 690-mile-long oil pipeline.

Taimur, 45, was appointed chief minister of Assam by Gandhi in December. She has used a silken rod to gain control of security in the state without further inflaming passions. Besides imposing censorship, she has deployed federal police to keep order -- leaving both the local police and the Army in the background.

She also ordered the arrest last week under preventive-detention laws of a deputy inspector general of the Assam police accused of working with extremist elements in the state.

Most observers believe the Gandhi government was able to wear down the movement, which picked up steam in the fall of 1979 over the issue of registering residents whom the Assamese considered foreigners. Many of them were illegal immigrants who had migrated over the porous border from Bangladesh as long as 33 years ago, while others were Indians from the densely populated neighboring Indian state of West Bengal in search of room to farm.

The government wants to allow all illegal immigrants who came in before 1971 to remain, although the would not be citizens, while the students have demanded the expulsion of those who entered after 1961.

It now looks, however, as if the students might accept the 1971 cutoff date.

For its part, the government has intensified its efforts to find and deport new illegal immigrants. Border security has been tightened and there are reports of as many as 3,200 foreigners being deported.

While the movement has mainly been aimed at keeping Assam for Assamese, who are culturally and ethnically different from other Indians, there were murmurs a year ago of secessionist tendencies.

The unrest in Assam fueled problems throughout all seven of India's northeast states, which have always felt cut off from the nation's heartland.

At its peak last spring and summer the Assam unrest had turned into India's largest mass demonstrations since the "quit India movement" that ended with his country gaining independence from Britain in 1947.

There was so much popular support for the rallies that government attempts to quell them in mid-April were met by massive protests with tens of thousands of Assamese defying curfew orders until they were rescinded. t

Now, however, the months of demonstrations have badly damaged Assam's economy. Both state and central government employes, whose support provided the backbone for the movement, were docked pay for 20 to 70 days' work. At the same time, the blockade caused shortages of food-stuffs and other essential commodities, bringing about price increases.

Most important to the rest of India, however, was the oil. Asssam provides about 20 percent of the oil India consumes and Assamese crude is vital for the production of fertilizer. Without the oil from Assam, the government was forced to spend an extra $1.375 billion to buy foreign oil at a time when its foreign reserves were dwindling.