Chasing 196 for victory at Hammond Field, the Young Windies motored to their target in the 29th over with the right-handed Kallicharan top-scoring with an unbeaten 87 off 57 deliveries.
Opener Kegan Simmons supported with 45 while Jeavor Royal ended unbeaten on 33.
Thula Ngcobo (2-26) and Mlungiseleli Gcado (2-48) finished with two wickets apiece.
Bishop had earlier claimed six for 17 from six overs as the hosts reached 194 for nine off their 30 overs after opting to bat first.
Opener W Rahman top-scored with 80 from 66 balls with seven fours and four sixes while Joel Veeran made 35 and Jason Wagner, 32.
Rahman added 56 for the first wicket with Sithabiso Zungu (10) before putting on a further 55 for the fourth wicket with Veeran.
When Rahman was dismissed in the 20th over, Veeran and Wagner came together to post 42 for the fifth wicket before Bishop helped run through the tail.
In reply, the Windies Under-19s were struggling on 52 for three in the 13th over before Kallicharan arrived to blast them to victory.
He struck five fours and six sixes and posted 49 for the fourth wicket with Simmons who faced 59 balls counted four boundaries.
When the Caribbean side lost Simmons and Alick Athanaze (4) in the space of 14 deliveries, Kallicharan combined with Royal in an unbroken 73-run sixth wicket partnership to see their side home.
Royal faced a mere 18 balls and struck four fours and a six.
West Indies Under-19s take on South Africa Under-19s in the first of five Youth One-Day Internationals on Sunday at Chatsworth Oval.