WASHINGTON — Democrats counted on solidifying their thin leadership grip on the Senate as voters flocked to the polls Tuesday to fill 35 Senate seats. Embattled Republicans braced for losses but hoped they would be modest ones.

Democrats were optimistic that the only two Democratic senators who will lose their Senate seats as a result of the national elections will be Barack Obama and Joe Biden.

And if the Democratic presidential ticket prevails, Democratic governors in Illinois and Delaware are sure to appoint Democrats to replace them.

Polls were open before daybreak in about a dozen Eastern seaboard states. Long lines appeared in many places where polls opened at 6 a.m. EST, and New Jersey Gov. Jon S. Corzine was seen reading a newspaper while waiting to vote at his Hoboken precinct.

Biden is running for re-election as senator from Delaware as well as for vice president. The Senate seats of Obama and GOP presidential candidate John McCain are not on the ballot.

Senate Democrats now have a tenuous 51-49 majority, and only thanks to the support of two independents. But a slumping economy, an unpopular war and voter fatigue after eight years of President Bush could help them bolster that majority, building on the six seats they added in 2006.

Piggybacking on aggressive Obama voter-registration and get-out-the-vote drives, especially in battleground states, Democrats’ goal Tuesday is reaching a coveted 60-seat, filibuster-proof Senate majority. Leaders in both parties portrayed that as a long shot.

Yet even bringing their numbers to close to 60 would enable Democrats to exercise far more control than they have now, since some Republicans probably would join them in efforts to break Senate logjams on many bills and judicial appointments.

Democrats had fewer seats to defend than Republicans. Of the 35 races on Tuesday’s ballot, 22 are now held by Republicans, 13 by Democrats.

The Democrats’ top prospects for pickups were those of three retiring GOP senators in Virginia, Colorado and New Mexico.

In Virginia, former Democratic Gov. Mark Warner faced another former governor, Republican Jim Gilmore, for the seat of retiring Republican Sen. John Warner. In Colorado, Democratic Rep. Mark Udall, son of the late Arizona Rep. Morris “Mo” Udall, faced former Republican Rep. Bob Schaffer for the seat now held by Republican Wayne Allard. And in New Mexico, Democratic Rep. Tom Udall — a cousin of the Colorado Udall — faced Republican Rep. Steve Pearce to succeed retiring Sen. Pete Domenici.

The three Democrats were comfortably ahead in pre-election polls.

Republicans seeking re-election faced tight contests in five other states — Ted Stevens in Alaska, Elizabeth Dole in North Carolina, Norm Coleman in Minnesota, John Sununu in New Hampshire and Gordon Smith in Oregon.

One route to 60 for Democrats was to win these five states on top of Virginia, Colorado and New Mexico, plus pick up at least one of three other closely contested races in Georgia, Kentucky and Mississippi. And not lose any Democratic seats.

Going into the election, only one incumbent Democrat appeared vulnerable: Sen. Mary Landrieu of Louisiana. Republicans made a spirited run at her, but Democrats saw her seat as a safe one.

In one of the most closely watched races, Alaska’s Stevens, at 84, the longest serving Republican in Senate history, sought re-election despite calls from GOP leaders to resign after he was convicted last week of seven counts of lying on Senate financial disclosure forms. He was locked in a tight contest with Democrat Mark Begich, the mayor of Anchorage.

Another closely contested race was in Minnesota, where Republican incumbent Coleman was challenged by Democrat Al Franken, the former “Saturday Night Live” writer and actor. A significant third-party candidate, Independent Dean Barkley, was complicating the race.

Underscoring the closeness of the race, Coleman embarked on an all-night bus tour with overnight stops in St. Cloud, Brainerd, North Branch, and Forest Lake before voting at 9 a.m. CST at the Linwood Recreation Center in St. Paul.

Democrats also took sharp aim at Senate Minority Leader Mitch McConnell, challenged by Democratic businessman Bruce Lunsford in a close and costly race in Kentucky. In North Carolina, a southern state into which the Obama poured heavy resources, Dole, the incumbent, was battling an aggressive challenge by Democratic newcomer Kay Hagan.

Sen. John Ensign, R-Nev., head of the party’s senatorial campaign committee, acknowledged ahead of the voting that “Democrats are poised to pick up some seats.” His Democratic counterpart, Sen. Charles Schumer, D-N.Y., predicted “a whole lot of seats” for Democrats, but said reaching a 60-vote majority was unlikely.