Osvaldo Alonso opened the scoring on 28 minutes, however Sunny Obayan found the net from a corner to level proceedings just before the break.
The hosts were heading for their second draw of the season before Olave got to the ball ahead of the oncoming keeper for the decisive goal of the game.
Visitors Started The Better
The Sounders came into the game looking for their first win of the season after a 1-0 defeat to Sporting Kansas City in the season opener, and it was the visitors who started the game brightly.
Seattle controlled most of the first-half and went close on a few occasions, with the closest coming from a bicycle-kick from Nelson Valdez on 12 minutes.
The visitors suffered set-back two minutes later when Erik Friberg, who had just returned to the line-up, was forced off after picking up a knee injury when challenging Jordan Allen.
Allen almost turned provider after coming on when he played the ball through to Joao Plata on 23 minutes. After taking a touch, Plata's shot was well saved by Tyler Miller, who got a glove to the effort, forcing a corner.
Sounders' Lead Cancelled Out Soon After
It wasn’t soon after that a chance eventually paid dividend when Nelson Valdez’s shot was parried into the path of Alonso, who calmly slotted home for his first of the season.
Chad Marshall’s powerful header on 34 minutes was parried away by Salt Lake keeper Nick Rimando, but after a neat break forward, the hosts were level.
The Royals made Seattle rue their missed chances as Sunny, who found himself in acres of space at the back post, headed home.
It was the hosts who could have gone ahead shortly after the restart, but after Chris Wingert drilled the ball across the box, Juan Manuel Martinez miss-kicked and the game remained level.
Substitute Darwin Jones, who replaced Jordan Morris because of a head injury, came the closest moments after coming on, but the chance went begging midway through the second-half.
Olave Ensures Royals Take All Three Points
The game looked to be heading for a draw, remaining knotted at 1-1, but Olave had other ideas and ensured the victory with a clever finish.
Martinez floated a cross into the box, and Olave managed to get ahead of the oncoming keeper, who came to claim the ball, and head home for the winner.
However, a second defeat of the season for Seattle means the next game, at home to the Vancouver Whitecaps on the same day, is crucial to get their season up-and-running.