The first 13 km stretch was inaugurated in June this year by Prime Minister Narendra Modi.
Both Mr Puri and Mr Vijayan rode on the second stretch spanning 5 km from Kaloor to Maharaja's College and back and then drove to the Town Hall for a public meeting.
Also present during the ride was "Metro Man" E Sreedharan, who was entrusted the job to see that the Kochi Metro becomes a reality. Besides, Leader of Opposition Ramesh Chennithala, local legislator Hibi Eden and Ernakulam Lok Sabha member KV Thomas were also present.
Like it happened in June, when Mr Vijayan named Mr Sreedharan, there was a huge round of applause.
With the 18 km stretch fully operational, it would take 32 minutes to cover the distance, that usually takes 90 minutes by road.
Kochi Metro authorities expect with the new stretch getting operational, the daily traffic would double and witness around 60,000 passengers.
The second phase of the project, stretching 7 km is also spread over two stretches. The work is progressing in the first stretch of 5 km.
The total length of Kochi Metro (till Tripunithura) including both phases is 25 km. The second phase is expected to be ready in another three years' time.
The work on Kochi Metro began in 2012 after the former Oommen Chandy-led Kerala government entrusted the project to Mr Sreedharan.