The woman (41) was arrested when gardaí stopped her after concerned passers-by saw her and raised the alarm, a court heard.
The catering worker had sought "comfort in alcohol" when she suffered depression after the birth.
Judge Bryan Smyth adjourned the case at Dublin District Court for the production of a probation report.
The accused pleaded guilty to being drunk while in charge of a child in Dublin city centre on December 24 last year.
Garda Sergeant Maire McDevitt said gardaí were alerted at 3.30pm by concerned members of the public who said a woman who appeared to be in an intoxicated state was pushing a buggy with a young child in it.
When the gardaí met the accused, she was staggering as she walked, her speech was slurred and she smelled strongly of alcohol. She was with her son.
The Childcare Act was invoked and the woman was arrested. She had no previous convictions.
The woman was married but the relationship broke down in 2019 and she had alcohol and mental health difficulties, her solicitor Tony Collier said.
At the time, her partner had the "lion's share" of access to their child and the woman now had one hour per week supervised, the court heard.
She had had a particular difficulty with depression after the birth and sought comfort in alcohol.
The accused had been off drink since the event which happened at a difficult time for the family.
"This has been a catalyst for change for her," Mr Collier said.
The welfare of the child was not central on that day but "it's central now and she wants to rebuild her relationship with the child and have greater access in the future", Mr Collier said.
She wanted to have overnights, a normal parenting relationship and get back to co-parenting with her partner.
It was a "very difficult situation", Mr Collier added.
The woman worked in the catering and hospitality sector and was living in private rented accommodation.
Judge Smyth said it was a "very serious matter" and sought a probation report.