Winning the first set 7-6 (2) in a tiebreaker, the Canadian teenager then bettered the performance and won the next set 6-4, but fell 7-6 (6) in the third and then 6-4 in the fourth.
Shapovalov, however, bounced back in a big in the final set and secured the win in what was a marathon clash that lasted nearly four hours.
Earlier, Ramkumar Ramanathan won the first match of the Davis Cup World Group Play-offs.
Ramanathan, ranked 154, defeated debutant Brayden SCHNUR in a marathon four setter.
Canada and India are now tied 1-1 after Day 1. Daniel Nestor and Vasek Pospisil will be in doubles action on Saturday against Rohan Bopanna and Purav Raja....