The study of 1,062 adult smokers found that a quarter of adult smokers also think the authorities should do far more to educate consumers about the minimum age, and highlight the requirement for proof of age when purchasing tobacco.
The survey was timed to coincide with the third anniversary of the raising of the legal minimum age for buying tobacco products from 16 to 18 years of age.
Other key findings in the survey included the fact that only 50% of smokers were aware of CitizenCard and No ID No Sale, while 24% still thought the 'Prove It' scheme existed.
However, 96% said they were aware that retailers asked for proof of age at the counter.
TMA chief executive Christopher Ogden said it was clear that more needed to be done, but that raising awareness of the legal age requirement was only part of the challenge. Tackling the illicit tobacco trade was also crucial, he added.
"Criminals operating in the illicit market show no concern for those they sell to and as a result many under the legal age gain access to illicit tobacco.
"In these times of austerity we hope that funding continues to be made available to crack down on this major source of lost revenue to the government."
Independent retailer Debbie Corris of Jim Ingram's in Whitstable, Kent told C-Store: "Just last week I was having a conversation with an adult customer who was unaware that the age limit had been raised.
"We made a big thing about it at the time, and we have all the correct posters and warnings on the tobacco gantry now, but clearly not everyone notices."