Elche, who avoided relegation on the final day of last term, took the lead through centre back Enzo Roco in the 11th minute after an exquisite through ball from Fidel Chaves.
Sevilla levelled in the 40th with a header from last season's top scorer Youssef En-Nesyri and went all out for the win late in the game.
New striker Rafa Mir had two good chances to snatch victory, skying a first-time strike from inside the area before heading over the bar with the final opportunity of the game deep in stoppage time.
"The point feels like very little," said Lopetegui.
"In the second half they didn't have a shot and barely came close to having one. We controlled the game, stopped our opponent playing as they wanted to but we couldn't score again.
"We played well enough and had enough chances to win."