Republicans looked set to hold at least 53 of the 100 senate seats, two more than they now have, and a slim majority of the 435-member house in the new 109th congress, which will allow them to wield greater power in the US capital.
Senate majority leader Bill Frist said it looked likely Republicans would win a total of 55 seats in the senate and add at least three seats in the house.
"It really is monumental. Nobody expected that. It is huge," Mr Frist told CNN, describing the election results as "a huge endorsement of the President of the United States".
Speaking on ABC, Mr Frist said he hoped strengthened Republican leadership in congress could help end the "extreme partisanship" that has plagued congress in the past few years.
However, Republicans will not have the 60 senators needed to end Democratic procedural hurdles against what critics have called "extreme" initiatives or nominees.
House majority leader Tom DeLay, a Texas Republican, predicted his party, with 227 house seats, would add at least a few more by the time vote counting ended.
"With a bigger majority, we can do even more exciting things," he told a local TV station in Texas.
A Republican Congress will help Bush push what promises to be a stepped-up conservative agenda, certain to include more tax cuts and anti-abortion judicial nominees.
Mr Bush will also make several nominations to the US Supreme Court, whose members must be confirmed by the senate.
Mr Daschle, the senate minority leader, was the Republicans' top congressional target and deemed the "chief obstructionist" to Mr Bush's conservative agenda. He was the first senate leader to be defeated since 1952.
Mr Daschle lost to former Republican John Thune.
All the house seats were up for re-election along with 34 senate seats.
But only nine of the senate races and about 30 of the house races were seen as competitive.
Republicans picked up previously Democratic-held senate seats in South Dakota, South Carolina, North Carolina and Louisiana, swapped a pair of other seats with Democrats in Illinois and Georgia and held hotly contested Republican senate seats in Oklahoma and Kentucky.
In Colorado, Democrat state attorney general Ken Salazar defeated brewing magnate Pete Coors for the seat being vacated by retiring Republican Ben Nighthorse Campbell.
In many cases, Republicans enjoyed the advantages of incumbency in fund raising and name recognition.
They also got a break in the senate races since many of the close contests were in largely conservative states where Mr Bush ran well.