Spain were producing a commanding display to assert their status as one of the pre-tournament favourites prior to De Gea's out-of-character error in Villarreal.
Vladimir Petkovic's determined Swiss outfit had kept four consecutive clean sheets but they were undone by a goal out of the top drawer from Alvaro Odriozola in the 29th minute.
David Silva's cross found the right-back, who advanced on to the high ball and cushioned a strike on the volley that swerved out of the reach of Switzerland goalkeeper Yann Sommer.
But when opposite number De Gea fumbled a weak shot from Swiss captain Lichtsteiner in the 62nd minute, fellow full-back Rodriguez could hardly miss as he arrived at the far post.
Sommer had to make a fine save low to his right to keep out a deflected Thiago Alcantara drive in the second minute as Spain started in similar form to that shown in their 6-1 demolition of Argentina in March.
Diego Costa's clever headed pass then provided Silva with a clear sight of goal, but the Manchester City midfielder sent a drive wide of the far post.
With Dani Carvajal battling to recover from injury, Odriozola was handed his third Spain cap and the 22-year-old opened his international account with a superb strike.
When Silva hooked a looping ball towards Odriozola on the edge of the box, the right-back sweetly struck a low volley that left Sommer with no chance.
Vamooooooos!!!! 1-0 grande @alvaroodriozola— Dani Carvajal Ramos (@DaniCarvajal92) June 3, 2018
Costa should have doubled Spain's lead in the 40th minute, misjudging a diving header from Koke's left-wing corner, Iniesta then jinking into the box but being denied by the busy Sommer.
Spain had an injury scare early in the second half as Iniesta took a bang to the knee in a challenge with substitute Breel Embolo, but the former Barcelona captain was able to play on before departing to a standing ovation at Estadio de la Ceramica.
Jordi Alba lashed a volley wide as Spain sought the cushion their dominant performance deserved, but out of nowhere Switzerland - two places above Spain in FIFA's world rankings - grabbed an equaliser.
Lichtsteiner's shot had little power but De Gea appeared to lose the ball in its flight, allowing it to slip out of his grasp, presenting Rodriguez with a simple close-range finish.
Nacho Fernandez clipped the outside of the post with a late opportunity to restore the Spain lead, while Switzerland rarely looked likely to repeat their 1-0 win at the 2010 World Cup in the last meeting between these sides.
Julen Lopetegui remains unbeaten in his two-year spell in charge of Spain and he will be able to call upon returning Real Madrid stars in Russia, but De Gea's unexpected error gave the coach something to ponder.