The Welsh region face unbeaten Ulster in Belfast on Friday following defeats by Edinburgh and Munster.
Owens is frustrated by the poor start, and acknowledged Ulster will be a tough challenge on their home ground.
"Ravenhill is probably the last place you want to be going when you've lost two on the bounce," he said.
"The pressure mounts, doesn't it?
"There's always going to be pressure but we're not hitting the panic button yet.
"It's the second game into the season and we could win the next 20 and we finish top!"
Scarlets' disappointing start follows a summer when they recruited Wales and Lions centre Jonathan Davies and Blues fly-half Rhys Patchell.
They were the top-placed Welsh region in the Pro12 last season, and are Wales' only representatives in the European Champions' Cup this term but have scored only one try in two league matches.
Coach Wayne Pivac says much of the early-season damage is self-inflicted.
"We're our own worst enemies," said the New Zealander.
"We're hurting ourselves at the moment more than the opposition are. They're taking advantage of our mistakes and we're nought from two as a result.
"Some of the statistics were in our favour - better than the stats when we were winning matches last season.
"But I'm disappointed that we're not converting opportunities two weeks in a row."
Captain Ken Owens says the Scarlets will not be hitting the panic button after losing their opening two games of the Pro12 season.