Ms Bacik topped the poll on the first count on Friday, and is poised for a significant victory for herself and her party.
Ms Bacik received a total of 8,131 first preference votes, taking a 30% share.
Arriving at the RDS on Friday, she said: “I’m overwhelmed, I’m over the moon. Certainly we were getting an enormously positive response on doorsteps across every area in the constituency.
“It perhaps reflects what we were hearing back, but of course you never know until you see the votes cast.”
Ms Bacik said her message that change is needed helped propel her to victory.
“What I’ve been hearing throughout the campaign was a message about the need for change,” she said.
She added: “From the non-political party campaigns I’ve been involved with for many years, I think that’s really what gave us an edge, that outpouring of support.”
It's a good day for us, for Labour, and all who adhere to Labour values of equality, of fairness and of solidarity
Ms Bacik acknowledged her victory was a significant one for the Labour party, which has struggled in the polls since their last spell in power ended in 2016.
She said: “I’ve been a Labour member since I was a student. I think people knew my record and my long history with the Labour party.
“I think it was a resounding message, the need for change, the need for a new voice and a voice that would represent this constituency, a centre-left voice.
“It’s a good day for us, for Labour, and all who adhere to Labour values of equality, of fairness and of solidarity.”
The count got under way at 9am and within a few hours Ms Bacik was pulling ahead of her biggest rival, Fine Gael’s James Geoghegan.
Labour members expressed hopes of victory, with many saying she is in a strong position to win.
Mr Geoghegan took 7,052 votes and Sinn Fein’s Lynn Boylan came in third place with 4,245 votes.
Following three counts, Ms Bacik is pulling ahead of the Fine Gael candidate by more than 1,000 votes.
Labour leader Alan Kelly said he is “very proud” of the way she performed in the by-election.
Speaking outside the RDS in south Dublin, he said: “Ivana is somebody who we have admired for many years, she has been a member with us all her adult life.
“Hopefully today will be her day and she will be elected to Dail Eireann.
“She put in an incredible performance. She was literally out morning, noon and night with a fantastic campaign team. It was a positive campaign. We are very proud of her.
“It’s a good day for our party.”
Polling in the contest to replace Fine Gael TD and former minister Eoghan Murphy closed at 10.30pm on Thursday, with the count taking place at the RDS.
Labour TD Duncan Smith and Ms Bacik’s campaign manager said he is “cautiously optimistic” that she will take a seat in the 33rd Dail.
Sinn Fein president Mary Lou McDonald said it is clear from the results that it is “Ivana’s day”.
Ms McDonald also said the Government is living on “borrowed time”, adding that Ms Boylan performed “very, very well”.
“Bear in mind there was a time, and it’s not a long time ago, the idea of Sinn Fein having a seat in Dublin Bay South would have been unthinkable,” she said.
“Now we have demonstrated that our base is very solid.
“We still have huge prospects for growth. I think we are in very good shape, I think our vote is good across the state but we are not complacent and each election is different.
“When you see what has happened to the Fianna Fail vote in particular, but to the Government vote more generally, I think it is safe to say that the Government has been given a very strong message from this constituency.
“I think they are on borrowed time.”
Meanwhile, Fianna Fail TD Jim O’Callaghan has cast doubt on Micheal Martin’s leadership following the party’s disastrous result in the by-election.
Asked if the Taoiseach should lead Fianna Fail into the next election, were it to go ahead as planned in 2025, Mr O’Callaghan replied: “We’ll have to think about that.”
Mr O’Callaghan, a TD for Dublin Bay South and director of elections for party candidate Deirdre Conroy, was speaking after Ms Conroy polled just 1,247 first preference votes.
Asked if he is concerned for his own seat at the next general election, he replied: “Certainly if the result is similar to this there will be more than faint alarm bells and I would have thought there will be alarm bells ringing in the heads of most Fianna Fail TDs in Dublin.
“Although this has been extremely disappointing and beyond what we thought was going to happen, there has been an awareness in Fianna Fail since the last election that the party has been declining nationally and in Dublin in the polls.”
Mr O’Callaghan said he would accept his share of responsibility for the result, but added: “I’m not exclusively responsible for Fianna Fail’s decline in the vote from 14% to 5%.”
Ireland uses a system of proportional representation rather than the first-past-the-post election method (FPTP) used in the US and the UK.
The proportional representation with a single transferrable vote system, referred to as PR-STV or simply PR, allows voters to rank candidates in order of preference.
If a voter’s first-choice candidate is eliminated, their vote is transferred to their next preferred candidate.
The idea is that it maximises a person’s vote, with fewer votes discarded than in the FPTP system.
Thursday’s vote was Ireland’s first electoral contest since the coronavirus pandemic began, but it remains to be seen if that affected turnout.
University College Dublin professor and anti-lockdown campaigner Dolores Cahill was prevented from entering the count centre after she refused to wear a face mask.
Ms Cahill, who ran as an independent candidate, attempted to push through a number of gardai and security officers to gain access.
An outspoken critic of Covid-19 restrictions, she was joined by a few supporters, who also refused to wear face coverings as required under health regulations.
The academic, who is no longer lecturing at UCD, demanded the identity of the garda, who repeatedly told her his name and the station he is associated with.
She left a short time later.
Other candidates in the election include councillor Claire Byrne, who is running for the Green Party.
The Social Democrats are represented by Sarah Durcan, while the People Before Profit candidate is Brigid Purcell.
The current TDs in the constituency are Fianna Fail’s Jim O’Callaghan, Sinn Fein’s Chris Andrews and Green Party leader Eamon Ryan.
Long considered a Fine Gael heartland, Dublin Bay South is home to the affluent suburbs of Terenure, Rathmines, Rathfarnham and Ballsbridge.
The other candidates in Dublin Bay South are Justin Barrett (National Party), Jacqui Gilbourne (Renua), Mairead Toibin (Aontu) and independents Peter Dooley, Mannix Flynn, John Keigher and Colm O’Keefe.