Two months after inconclusive national elections, Christian Democratic Premier-designate Francesco Cossiga succeeded today in forming a new government, Italy's 42nd since the fall of fascism.

A former interior minister who won widespread respect for his hard-line handling of the 1978 Aldo Moro kidnap case, Cossiga, 51, was the fourth politican since June to try to form a government capable of winning a majority in parliament. Cossiga took responsibility for the government stand in the Moro case and resigned after his death.

The new, technocrat-dominated Cabinet will be sworn in Monday and is expected to win a vote of confidence later in the week, in time for the politicians to take advantage of what is left of Italy's traditional August summer holiday. It will then become the first fully constituted government here in more than seven months.

Cossiga's predominantly Christian Democratic 24-member Cabinet also includes representatives of two smaller parties - the Social Democrats and the right-of-center Liberals - as well as two economic experts considered close to the Socialist Party.

To win the Socialists' backing, essential for a majority, Cossiga said the new government is not a genuine coalition. Last week the Socialists refused to support a government proposed by caretaker Treasury Minister Filippo Maria Pandolfi - who retains that post in the new government - on the ground that it was a copy of the three-party government headed by Giulio Andreotti that took the country into the June election.

The Socialist Party, Italy's third largest, made only minimal gains in those elections to win 9.8 percent of the vote and 62 seats in the 630-member Chamber of Deputies. But the powerful Communist Party's decision to end a three-year policy of support for minority Christian Democratic governments gives the smaller Marxist party the balance of power.

Cossiga can count on 291 votes in parliament with his partners plus a probable 16 votes from the Republicans. Socialist votes, or absentions, are thus needed.

With the Communists taking to the sidelines, the key to Italian politics has been a deadlock between the Christian Democrats, with 38 percent of the vote, and the Socialists, once partners in a 15-year center-left experiment. After a month of cross-vetoes that nullified attempts to form governments both by Andreotti and by Socialist chief Bettino Craxi, the two parties agreed to a temporary truce.