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Chasing a modest 186, India were in doldrums at 5 for 68 before Pujara played the captain`s role exceptionally well to guide the visitors to a nerve-wrecking win.
Pujara added 73 runs with number 10 batsman Shami Ahmed (27) to steer the Indians home after a spectacular collapse had severely jeopardised the team`s chances. Pujara, who has struck a 50 in the first innings, was once again patience personified as he held one end even as wickets tumbled at regular interval around him.
The 24-year-old right-hander faced 222 deliveries and struck 10 fours in a calculated innings that should impress the selectors when the senior squad is picked.
For the home team, lanky pacer Jason Holder, playing at his home ground, registered his third five-wicket haul in his 15th first-class match -- a performance that reduced India A to 115 for 8 at one stage.
On a rain-interrupted day, a draw seemed likely when the two teams went off the field in the final hour with India `A` needing 20 runs but the weather cleared and Pujara ensured that it was an opportunity grabbed as he took his side to a nail-biting triumph.
When rain struck the match for the first time, at lunch break, West Indies `A` pacers had added three more wickets to the overnight 22/3, giving away just 55 runs in the process.
Most of India`s batting was a shambolic effort with six of their willow-wielders falling to single-digit scores.
In fact, Ahmed`s 27 was the next best score on the sheet after Pujara`s momentous 96.
Rohit Sharma, who had rescued the team with a 94 in the first innings, battled hard for a 60-ball 23, but could not build on the patient effort -- dismissed by pacer Derlorn Johnson.
Rain during the lunch caused a 45-minute delay in resumption and it only strengthened Pujara`s resolve, who was at 26 before the break.
With wickets falling around him, Pujara shifted gears in the later half of the chase and became more aggressive.
India `A` were 115 when Holder dismissed number nine Bhuvneshwar Kumar (3), giving the hosts a good chance of sealing a win.
But Pujara and Ahmed added more than 50 before heavens opened up again to cause a 15-minute interruption.
India `A` needed 20 at that stage and after resumption, Pujara eased the pressure with a four before finishing it off with another boundary in the end.
The second and third match of series will be played at Kingstown and Gros Islet respectively.