A previous ruling on slaughter caused controversy
The bill, passed by a huge majority after more than a decade of debate, includes animals in a clause obliging the state to respect and protect the dignity of humans.
The upper house of parliament, the Bundesrat, is likely to approve the bill when it considers it next month, thus making Germany the first country in Europe to give constitutional rights to animals.
German law already defines the conditions under which animals can be held in captivity.
Ms Kuenast described the change as "ground-breaking", but said it would not diminish human rights.
"People remain the most important", she said.
Vets' organisations welcomed the vote, but the Society for Health and Research called the change Black Friday, and said it would lead to legal insecurity in research and education.
Damaging ruling
If adopted, the changed article in the constitution (or Basic Law) will read: "The state takes responsibility for protecting the natural foundations of life and animals in the interest of future generations."
Conservative opposition parties have for years fought efforts by environmentalists to introduce the amendment, saying it would tie Germany's hands in scientific research and lead to a brain drain.
But a widely criticised ruling by the constitutional court in January, authorising the traditional Islamic slaughter of animals without use of anaesthetic, lent new momentum to the animal rights movement.
The court had ruled that religious freedoms were explicitly protected under the Basic Law while animal rights were not.
All parties also recognise the need to rebuild confidence in the food and agriculture industry following a series of crises, from BSE to foot-and-mouth disease, says the BBC's Rob Broomby.
Better conditions for animals are seen as part of that agenda.