The Canadian government, accepting American assurances that the joint Canadian-U.S. Alaska natural gas pipeline eventually will be completed, today granted approval for construction of an initial section. It will bring surplus Canadian natural gas to American cities as early as next year.

The decision -- subject to routine approval by the National Enegry Board -- will allow work to begin this summer on the long-delayed $20 billion project and bring to an end more than two years of deliberations on the highway pipeline's viability. The uncertainty surrounding it had become a recurring irritant in relations between Washington and Ottawa.

Announcing the decision tonight, a Canadian government spokesman revealed a letter received today by Prime Minister Pierre Trudeau from President Carter reiterating U.S. pledges that the entire pipeline will be completed.

Saying that "the expeditious construction of the project remains in the mutual interests of both our countries," Carter said in the letter, "I would be prepared at the appropriate time to initiate action before the U.S. Congress to remove any impediment as may exist under present law to providing that desired confidence for the Canadian portion of the line.

"Accordingly, I will take appropriate action directed at meeting the objective of completing the project by the end of 1985."

Since 1977, when Canada and the United States agreed to build the gas pipeline, the question of financing, particularly for the expensive northern sections, has been an issue before American and Canadian regulatory bodies.

Even though financing for the whole pipeline could not be assured, Canada has been under pressure from the project's sponsors and from the U.S. government to build an initial section of the line. That section would be used to export surplus gas in the western Canadian provinces of Alberta and British Columbia to the United States.

The project's sponsors -- Northwest Alaskan Pipeline Co., of Salt Lake City and Foothills Pipelines Ltd. of Calgary, Alberta -- have argued that prebuilding the initial section of the pipeline in southern Canada would provide revenues that would contribute to funding construction of the rest of the pipeline at a later date.

As originally envisioned, the pipeline was to carry only Alaskan natural gas through Canada to the U.S. market. From Prudhoe Bay on Alaska's north slope, the pipeline would carry gas for 4,794 miles along the Alaska Highway through northern Canada to Alberta. There, the line would split into two branches, one leading to San Francisco and the other to the Chicago area.

Canadians worried, however, that the northern sections of the pipeline to tap Alaska gas might not be built for many years, if at all, if the lower portion of the line were completed.