Figures released through Freedom of Information reveal a big increase from just eight in 2011-12, 13 in 2012-13, and 21 in 2013-14.
Last year, the number of such babies rose to 34, with a further 15 instances recorded so far in the 2015-16 financial year.
Labour councillor Kevin Keenan said: “The real concern here is for the children who did not make a choice and risk growing up with the sins of their parents.
“We need to work with medical professionals to tackle the drug problem and give people the support they need to bring this figure down.”
All babies born to drug-dependent mothers in the area are treated in NHS Tayside’s postnatal ward or, if necessary, referred to the neonatal intensive care unit.
Not all of the babies recorded in the statistics underwent a drug treatment programme and the figures include those who were discharged without treatment.
Earlier this year, in an Early Development Journal study, 80 babies born to drug-addict mothers were found to have significant developmental and vision problems at six months old. The Scottish study found they performed less well on tests for social interaction, co-ordination and motor skills.
A Scottish Government spokeswoman said: “The number of adults reporting drug use fell by 1.4 per cent between 2008-9 and 2012-13 and is continuing to fall, while drug taking among young people is at its lowest level in a decade.
“It is crucial that pregnant women and their unborn children get access to high-quality healthcare as we know the important contribution it makes to healthy lives going forward. This is especially relevant when the mothers have drug dependencies.
“Since 2007, the Scottish Government has dramatically reduced drug treatment waiting times and provided specialist treatment and recovery to over 40,000 individuals, some of whom are expectant mothers with drug dependencies.
“We have also invested over £194 million in front-line drug treatment services.
“Alongside this, we’re also actively participating with the UK government in a UK-wide review of the clinical guidelines that support clinical decisions for those with problem drug use, where particular consideration is given to pregnant women.”