(CNN) Karen Handel's victory in the Georgia special House election Tuesday night has House Republicans who recently voted for a deeply unpopular bill to repeal Obamacare breathing a sigh of relief.

And it will now offer some cover -- even political reassurance -- to Senate Republicans who are gearing up to cast similar votes next week.

The Democrat in the Georgia race, Jon Ossoff , was unsuccessful in flipping a traditionally Republican district in the Atlanta suburbs previously represented by Health and Human Services Secretary Tom Price. Had the 30-year-old first-time political candidate pulled off an upset, it would have dealt a major blow to the Republican Party's already complicated efforts to gut Obamacare.

Democrats were prepared to cast the results of the closely watched special election as a referendum on President Donald Trump and the GOP's legislative priorities -- chief among them the quest to repeal former President Barack Obama's landmark health care law.

Former Republican Minnesota Gov. Tim Pawlenty said Handel's win provides the party with "huge relief and somewhat of a political sedative" heading into 2018, when Democrats are eager to try to win back control of the House.

"It kind of calms the waters in terms of people looking for predictors or harbingers and what it means for 2018," Pawlenty told CNN. "It says: Perhaps the approach that's being taken in Congress and by the President are more acceptable to a swing district or swing-voting parts of the country than people are predicting."

Since House Republicans narrowly passed a bill to repeal Obamacare in May , Ossoff was increasingly outspoken in his criticism of the legislation.

JUST WATCHED GOP did not mention Trump after Georgia win Replay More Videos ... MUST WATCH GOP did not mention Trump after Georgia win 00:47

The House bill would dismantle the main pillars of the current health care system, eliminating Obamacare taxes on the wealthy and insurers; getting rid of the individual mandate; significantly curtailing federal support for Medicaid; and allowing states to loosen key Obamacare protections for people with pre-existing conditions

The controversial proposal, which Handel said she would have voted for, would "gut the protections for Americans with pre-existing conditions -- hundreds of thousands of them," Ossoff argued at a debate earlier this month.

Handel pushed back forcefully, pointing out that her sister was born without an esophagus -- a pre-existing condition.

"For you to suggest that I would do anything to negatively effect her is absolutely outrageous and unacceptable," Handel said.

JUST WATCHED Jon Ossoff concedes Georgia race Replay More Videos ... MUST WATCH Jon Ossoff concedes Georgia race 01:37

With health care so much of the focus in the Georgia special election, Democrats were ready to liken an Ossoff victory to that of former GOP Sen. Scott Brown in the 2010 special election in Massachusetts.

At the time, Brown's unlikely triumph over Democratic Attorney General Martha Coakley was viewed as voters' sharp rejection of Obamacare, which Democratic lawmakers were in the middle of crafting. When Brown decisively won the office long occupied by the late-Democratic Sen. Ted Kennedy, Democrats were forced to act quickly to pass Obamacare, despite deep reservations and divisions across the party about the legislation.

Now, with Handel keeping Price's old seat in GOP hands, Democratic strategists insist that health care will still be powerful ammunition against Republicans in next year's congressional elections.

"I don't think that very many Republicans will take much comfort on the health care issue even if Handel does win," said veteran Democratic pollster Geoff Garin in advance of Tuesday's result. "I think Republicans will continue to recognize that taking away coverage from millions of Americans and raising costs for millions more is a politically unpopular and dangerous enterprise."

A draft of Republicans' plans in the Senate is expected to be released this Thursday, according to Senate Majority Leader Mitch McConnell, who wants to put health care in the rear-view mirror before lawmakers leave Washington ahead of the July 4 recess.

Senate Republicans have already been facing mounting pressure -- including from the President himself -- to address serious concerns that have been raised about the House GOP bill. Despite having celebrated the House passage of an Obamacare repeal bill last month, Trump privately told lawmakers recently that the House proposal was "mean," and that the final product plan must be more "generous."

White House spokesman Sean Spicer reiterated that sentiment in the briefing room Tuesday.

"The President clearly wants a bill that has heart in it," Spicer said. "He believes that health care is something that is near and dear to so many families and individuals."