Morgan opened scoring for the U.S. in the 16th minute with a heel flick off a pass from Mallory Pugh that got past Haiti goalkeeper Lara Larco. Morgan's second goal came on a header in the 23rd minute.
Monday's win at Estadio Universitario in Monterrey, Mexico, saw the USWNT overcome defensive struggles against the upstart Haitians and secure the first match at the inaugural tournament among the top sides of North and Central America and the Caribbean.
Roselord Borgella could have given Haiti a goal from the penalty spot after Morgan's two early goals, but she missed and then was later given a yellow card for a foul before the half.
The United States appeared to score on an own goal in the 78th, but Megan Rapinoe was ruled offside. Purce scored her fourth international goal in the 84th.
U.S. coach Vlatko Andonovski praised Morgan and said he was pleased with the result, although there's things to work on before the Americans face Jamaica on Thursday.
"On the other side, we gave up a lot of opportunities, gave up a penalty kick. That's something we're going to have to do better and tie up a few things for the next game,'' Andonovski said.
Eight teams are split into two groups for the W Championship. The top two finishers in each group earn berths for the 2023 Women's World Cup co-hosted by Australia and New Zealand. The third place finishers in each group advance to a 10-team intercontinental playoff.
The winner of the W Championship also earns a spot in the 2024 Olympics in Paris.
The Associated Press contributed to this report.