Exxon Corp., the biggest U.S. oil company, reported yesterday that profits rose 31.8 percent year from 1979 levels but fell 1.1 percent in the fourth quarter due to higher exploration costs and reduced petroleum demand.

Exxon said fourth-quarter net income totaled $1.35 billion ($3.12 a share) against $1.37 billion ($3.10) a year earlier. Sales rose to $29.85 billion from $25.82 billion.

Earnings in 1980 were $5.66 billion ($13.01), up 31.8 percent from the $4.295 billion ($9.74) earned in 1979. Sales rose to $110.46 billion from $84.97 billion, and return on shareholders' equity climbed to 23.6 percent from 20.1 percent.

Meanwhile, Marathon Oil Co., the 16th-largest U.S. refiner, said yesterday that it logged a 1.9 percent increase in profits.

Marathon earned $55.3 million (92 cents a share) compared with $54.3 million (90 cents) in the 1979 final quarter. Revenues were up 9 percent to $2.42 billion from $1.22 billion.

For the full year, Marathon's earnings were up 17 percent to $379 million ($6.27 a share) from $323.2 million ($5.34) in 1979. Revenues were up 22 percent to $8.8 billion from $7.2 billion in 1979.

Exxon's worldwide capital and exploration expenditures totaled $8 billion in 1980, up 8.2 percent from the $7.4 billion of expenditures in the previous year. Capital and exploration expenditures in the United States accounted for 44 percent of the worldwide total.

Exxon said its U.S. refining and marketing operations lost $58 million in the fourth quarter, against a fourt-quarter loss of $72 million in 1979. But U.S. petroleum exploration and production earnings jumped 27 percent to $565 million because oil and natural gas prices were rising.

Exxon said helping to offset its fourth-quarter earnings decline was a $215 million gain on foreign-exchange translations, "up substantially" from the $2 million gain reported a year before.

Exxon's earnings were about in line with those of most other U.S. refiners, which have reported essentially flat earnings for the October-to-December period last year.

Industry analysts had anticipated fourth-quarter profit increases of 15 percent to 20 percent for Exxon and for Mobil Corp., Texaco Inc. and Standard Oil Co. of California, the three other U.S. partners in the Arabian American Oil Co., which produces the bulk of Saudi Arabia's oil. As Aramco partners, they have access to Saudi Arabia's relatively low-priced oil at $32 a barrel.