That Sri Rama worshipped Supreme Goddess is mentioned in Lalita Trishati Stotram found in the Brahmanda Purana:
Lakaararthaa Latha Poojya Laya Sthithudbhaveswari
Lasya Darshana
Santhushta Labhaalaabha Vivarjitha
Langhyetharaagjnaa Lavanya
Shalini Laghu Siddhida
LaakshaarasaSavarnaabha Lakshmanagraja
Poojitha
Labhyetara Labdha Bhakthi Sulabha Langalaayudhaa
Lagna Chaamara Hastha Shri Sharada Parivijitha
Lajjaapada
Samaraadhya Lampata Lakuleshwari
Labdha Maanaa Labdha Rasaa Labdha
Sampath Samunnatih ||
You are esteemed and
worshipped by the womanhood of chastity; the Supreme of the Universe
in all the stages of Birth, Growth and Destruction alike; witnessing
the joyous dance of women gives you immense pleasure; You are far
above the considerations of profits and losses; Mother, your
instructions are always carried out and never questioned and indeed
none whosoever dared to ask you questions! You are matchless in
elegance and poise; you give away Siddhis with least devotion and
reverence; you shower kindness with minimal efforts like melting gold
is sealed in wax; Lord Shri Rama the incarnation of Vishnu performed
regular pujaas to you to bestow the capacity to uproot Adharma; you
save persons regardless of their deeds and fruits; you are very easily
won over with selfless devotion; some times you carry plough as a
weapon like Adi Sesha to punish the vicious; Lakshmi and Sharada serve
you with Chaamara as a hand fan; you make exceptions to Principles of
your own making some times to save situations; Devi! Several Groups of
persons from varied backgrounds are unanimous in their unity in
devotion to you albeit in the methodologies of their worship; you are
venerated by one and all as you are the Finality of Happiness and
Prosperity.
Lakshmanaagraja pujita--- One who is worshipped by Lakshmana's elder brother referring to Sri Rama.
But, the full details of the story of Akala Bodhana (or the untimely invocation of Goddess) is found in the Kalika Purana (which is one of the 18 Upa-puranas).
I have read that in the introduction of the Text itself.
UPDATE:
More details of Sri Rama worshipping the Mother Goddess is found in the Devi Bhagavatam as well.
From the Purana's 3rd book's 28th chapter:
- Janamejaya said :-- O Muni! How did Râmchandra celebrate the Devî’s Pûjâ, that leads to happiness? Who was He! And how was stolen away His
Sîtâ? How was He deprived of His kingdom? Please satisfy me by
narrating all these incidents to me.
In the 29th and 30th chapters, the details are found.
It was Narada who described the methods of worshipping the Goddess to Sri Rama in the 29th chapter:
- O Descendant of Raghu! I am now telling how that Râvana can be killed. Perform, in this very month of Âs’vin, the vow with devotion.
- Fasting for nine nights, the worship of the Bhagavatî, and repeating the Mantram silently and performing the Homa ceremony,
observing all the rules, will certainly fulfill one’s all the desires.
- O the best in the race of Raghu! You should offer the sacrifice before the Goddess of a sacred and unblameable animal, perform Japam
and Homa ceremony equivalent to onetenth of Japam. If you do all this,
you will certainly be able to release Sîtâ.
In days of yore, Visnu, S’iva and Brahmâ and the Devas in the Heavens all performed this worship of the Goddess.
Therefore, O Râghava! Every person desiring happiness, specially those that have fallen under great difficulties, ought to do this
auspicious ceremony, without the least hesitation. 23-24. O Kâkutstha!
Vis’vamitra, Bhrigu, Vas’istha and Kas’yapa all of them did this
worship before. When some stole away the wife of Brihaspatî, the Guru
of the Devas, he, too, by the force of this worship, got his wife
back. Therefore O king! dost thou also celebrate the Pûjâ for the
destruction of Râvana.
Râma replied :-- “O Ocean of Knowledge! Who is that Devî? What is Her influence; whence has She sprung? What is Her Name? And how is
that vow to be duly observed? Kindly describe all these to me in
detail.”
- Nârada answered :-- “Listen, O Râghava! That Goddess is Eternal and Ever Constant Primordial Force. If you worship Her, all your
difficulties will be removed and all your desires will be fulfilled.
She is the source of Brahmâ, Visnu and others and of all these living beings. Without Her force, no body would be able even to move
their limbs.
Râma said :-- “O best of the Munis! Describe briefly all the rules and regulations as to how that vow and worship are to be performed.
With my heart, full of devotion and faith, I will worship the Goddess
today.
And, finally Sri Rama worshipped Mother Goddess and appeased her:
41-42. Vyâsa said :-- Then the powerful Bhagavân Hari, hearing all
from the Muni, believed them to be true; and, on the approach of the
month of Âs’vin, prepared the altar on the top of a hill and placed
the Auspicious Goddess, the World Mother and, observing all the rules,
performed the vow and worshipped the Goddess.
43. Fasting for nine days, Râma celebrated the vow and duly offered sacrifices, performed the worship and Homa ceremonies. 44-46. When, on
the grand night of the Eighth lunar day, the two brothers completed
the vow as told by Nârada, the Supreme Bhagavatî was pleased with the
worship and appeared before them, mounted on a lion, and remaining
there on the mountain top, addressed Râma and Laksmana, in a sweet
grave tone, like the rumbling of a rain cloud, thus :-- “Râma, I am
satisfied with your worship; ask from me what you desire.
47. Râma! You are sent by the gods for the destruction of Râvana and are born as a part incarnate of Nârâyana, in the pure and stainless
family of Manu.
