Republicans in Northern Virginia made gains in yesterday's elections, ousting Arlington's controversial prosecutor William S. Burroughs Jr., picking up three House seats in the legislature and adding a fourth Republican to the nine-member Fairfax County Board of Supervisors.

The region's burgeoning GOP also retained majority control of the Arlington County Board but failed in the six races in which it challenged Democrats in the state Senate.

Elsewhere in Virginia, GOP Senate candidates were more successful, de- feating three Democratic incumbents. One was in Roanoke and two were in Southside Virginia, where Del. Eva Scott, an independent turned Republican, defeated a Democratic senator to become the first woman in the 40-member state Senate. In the House, she led successful fights to block ratification of the Equal Rights Amendment.

In a major upset for Arlington Democrats, Republican-backed challenger Henry E. Hudson claimed the commonwealth's attorney post from Burroughs with 59 percent of the votes cast to Burroughs' 41 percent.

Hudson, once chief assistant to Burroughs, made much of the continuing controversy surrounding the prosecutor's handling of a celebrated 1977 double murder case involving Richard Lee Earman. Burroughs, the county prosecutor since 1974, declined through a campaign aide to comment on his loss.

Republicans added three new House seats in Fairfax County, defeating one freshman Democratic delegate and taking two other seats formerly held by Democrats. The victories cost the GOP the defeat of one of its most outspoken legislators, Del. Robert L. Thoburn, a fundamentalist minister.

Fairfax Republicans narrowed the longstanding Democratic majority on the Board of Supervisors from 6-to-3 to 5-to-4. A key race there was Republican newcomer Thomas M. Davis III's easy victory over Democrat Betsy W. Hinkle.

Republican-backed candidates retained their 3-to-2 majority on the Arlington Board as incumbents Dorothy T. Grotos and Walter L. Frankland Jr. defeated two Democratic-backed challengers.

Fairfax Board Chairman John F. Herrity won reelection by a wide margin over Democrat Vivian Watts, and Wayne Huggins, the county's chief deputy sheriff, beat out a Democrat and an independent to succeed his controversial mentor, four-term Sheriff James Swinson.

Clearly licking their wounds, Democrats who lost and even some who won cited several reasons last night to explain the results.

"We now have, in many cases, election by ambush where those who muster the strongest last-minute attack can win close elections," said State Sen. Adelard L. Brault (D-Fairfax), who fought off an aggressive and well-financed challenge by 22-year-old John M. Thoburn, a conservative Republican.

Brault, 70, who returns to his influential duties in Richmond as Senate majority leader, complained that John Thoburn, the college student sone of Del. Thoburn, had riddled his campaign with voting record distortions.

In an even closer battle for the Senate seat vacated by a retiring Democratic legislator, Del. Richard L. Saslaw kept that office in the party, defeating Republican challenger James R. Tate, a former delegate.

Two other Fairfax senators, Clive L. DuVal II and Joseph V. Gartlan Jr., both Democrats, were easily reelected as was State Sen. Charles L. Waddell (D-Loudoun).

Republican victories in the House were accomplished by winning two open seats in Fairfax County that incumbent Democrats had abandoned to run for higher office. Another Democrat, Kenneth R. Plum, lost to a Republican in his bid for reelection in the 18th District.

The GOP's significant gains in the region appeared to bolster the party's hopes that Republicans can whittle away at a legislature and local governments traditionally dominated by Democrats.

Republican Gov. John N. Dalton has made several campaign trips north on behalf of GOP candidates. In what Democrats labeled the most partisan politicking of any governor in memory, Dalton traveled around the state urging election of a vetro-proof assembly.

Although pulling back from such a sweeping goal in recent days, the GOP made no secret of its desire to duplicate its statewide gains at the local level.