Just before halftime the Englishman found himself in an ideal position in the box but frustratingly headed wide.
When Irishman Colm Kenny scored for Olympic to level, 1-1, it was their first decent chance of the game.
With coach Mick Waitt's inspired team talk, the hosts raised their game. A brace from Sam Blackburn and one from Jimmy Haidakis sealed victory.
Tinsley got Rovers back to 3-2, ironically after a dominant first period, but it was virtually their only chance in the second.
Waitt considered his team's first-half display the worst in his three years of coaching the side and had to steady himself before going into the dressing room at the interval.
"I was going to go ballistic, I must admit, but I changed my mind and tried to be more constructive about things," said the former All Whites coach.
Naturally, his opposite number was left wondering what could have been if the first half had turned out more favourably.
"In the first half we absolutely dominated the game and should have been 5-0 up at the break," Robertson said.
"We conceded one goal with their only attacking move, a set piece. In the second half they took their chances and hit us on the counterattack and were more clinical than we were."
Napier have defeated the two Hutt Cities (Lower and Upper) fairly comfortably but rode their luck against Tawa. They lost heavily to Miramar Rangers and now another title contender has exposed them.
However, Robertson brushed off suggestions they were unable to keep up with the Joneses.
"I don't think Olympic were a better side than us today. It was a simple case of us not taking our chances and that was the difference between us."
In other league matches on Saturday, Palmerston North Marist held Western Suburbs to 2-all and Tawa upset Lower Hutt City 3-1. Wairarapa United drew 2-2 with Upper Hutt while Petone's match against Miramar Rangers was postponed.