However, the Indian defence did a fine job to see out the rest of the time and secure a memorable win.
At the start, India began with an intent to hit Estonia on the counter, using their speedy wingers while letting the opponents drive deeper into the box.
This played into the hands of the Blue Tigresses from the start, as Karthika's interception on the right allowed Pyari to rush into the space behind the Estonian defence.
The India striker laced her shot past Estonia keeper Karina Kork, but it struck the upright and was eventually cleared.
Pyari often acted as the focal point of the counterattacks for India, and she created a great opportunity for the Blue Tigresses in the eighth minute when she made a swift turn to leave her marker stranded at the halfway line and played a through ball to Manisha on the left with acres of space.
However, the latter scuffed her shot.
Manisha made up for her earlier miss, however, when she initiated an India move with a clever back-heel to Pyari, who put Anju Tamang through behind the opposition defence.
A melee ensued inside the Estonia box as Anju's shot was blocked, but Manisha collected the ball and drove it in from the near post from a narrow angle.
India ceded more control of the ball after taking the lead and paid the price for it a little after the half-hour mark. Vlada Kubassova sent in an accurate cross from the left, which was nodded in by Tammik.
Chaoba Devi's side tightened the screws as they came out in the second half, enforcing a mid-block, which meant that Estonia could no longer control the proceedings in the middle of the park.
India took the lead a little after the hour mark through Indumathi.
Soumya Guguloth initiated the attack on the right, passing it to Anju, who squared it to Pyari, who, in turn, laid it off for Indumathi at the edge of the box. The India midfielder unleashed a left-footed shot that snuck into the bottom corner.
The Blue Tigresses began pressing harder, and that, along with some cleverness from Anju and a bit of desire and individual brilliance from Pyari, resulted in the third goal.
Soumya snatched the ball in the middle and played it through to Anju on the right. However, the latter was in an off-side position and kept herself as far away from the ball as possible.
Meanwhile, Pyari made a lung-bursting sprint to the right wing, to get there before any of the Estonia defenders. She barged into the box from the right, cut inside, and placed her left-footed effort into the bottom corner to double India's lead.
Manisha added a fourth, her second of the day, just a couple of minutes later, controlling a cross by Sandhiya Ranganathan, to score at the near post again.