The Croatians took the lead through a wonderful Leon Benko volley in the 10th minute but after Betis levelled through Mabwati Cedrick's rising drive four minutes later there was only going to be one winner, with the Spanish side dominating.
Betis failed to make further inroads mainly due to the heroics of Vargic, who offered a statement of intent midway through the first half with three fine saves in under two minutes as the home defence creaked under the visitors' pressure.
Rijeka impressed in defeating Stuttgart in the play-offs to earn their place in Group I, but were brought crashing down to earth on matchday one as Guimaraes hammered them 4-0.
Better was required on Thursday night and the hosts duly responded by taking the lead with their first attack.
Ivan Tomecak was involved in the move, cutting inside his marker down the right and flashing a drilled cross into the area for Benko, who athletically volleyed beyond the helpless Stephan Andersen.
The hosts were unable to maintain their lead for long, however, as Betis swarmed forward to apply pressure on the edge of the Rijeka box, where the ball eventually broke to Cedrick, who slammed a fierce shot into the top-right corner.
Betis were held to a goalless draw by Lyon in their opening group fixture but clearly fancied their chances of going on to win this one and moments later went close to adding to the scoring when Vargic kept out Salva Sevilla's low effort.
Even better was to come from the home goalkeeper midway through the half as he first tipped over Cedrick's powerful 25-yarder before clawing away Braian Rodriguez's shot from the resulting corner.
And from the next corner Vargic was at it again, getting across well to parry clear Juanfran's strike from the edge of the area.
Into the second half and the traffic remained all one-way, with Juanfran again bringing the best out of Vargic after an hour, while with 12 minutes left Cedrick headed the final chance of the match wide of the target.