WASHINGTON—President Barack Obama’s move to offer legal-worker status to several million undocumented immigrants will send unpredictable ripples through the U.S. economy, prompting many to seek higher-paying jobs and heightening competition for positions in a number of sectors, economists say.

Those studying the potential impact of the president’s executive order, which he announced on Thursday night, point to the Reagan-era 1986 Immigration Reform and Control Act, which allowed about 1.7 million undocumented immigrants to...