New Orleans, beaten by an overtime field goal in last season's NFC Championship game against the LA Rams, looked set for more disappointment as they trailed 14-3 and then 21-10 at the Mercedes-Benz Superdome.
The hosts rallied with 17 unanswered points to take the lead, but Houston hit back and quarterback Deshaun Watson's 37-yard touchdown pass to Kenny Stills with less than a minute left put them ahead again at 28-27.
However, victory was snatched away from the Texans when, following a rapid six-play drive from Drew Brees which put the Saints in field goal position, Lutz drilled over a long-range kick as time expired.
In Monday night's other match, the Oakland Raiders defeated the Denver Broncos 24-16.
Oakland got off to a flying start when Derek Carr threw a touchdown pass to Tyrell Williams off their opening drive of the season and the Raiders never looked back.
Josh Jacobs also ran in two touchdowns for Oakland as Denver's NFL record of seven successive opening-weekend wins came to an end.