Clark's unassisted goal came only 19 seconds after Sean Bergenheim had opened the scoring 11:15 into the first period for the Lightning, who stunned Boston with a one-sided road win in TD Garden.
Purcell made it 3-0 only 1:06 later on another unassisted tally, and the Bruins never recovered.
Marc-Andre Bergeron added a power-play goal and Simon Gagne contributed an empty-netter for the Lightning, who won their eighth straight post-season game despite a nine-day layoff since their last contest.
Dwayne Roloson turned aside 31-of-33 shots for the Bolts.
Rookie Tyler Seguin got his first career post-season goal and Johnny Boychuk added a tally for Boston, which will try to even the series in Tuesday's Game 2.
Tim Thomas stopped 29-of-32 shots in the Bruins' loss.