After starting eighth on the grid, Ricciardo completed a fine race to finish fourth behind Mercedes' winner Nico Rosberg, with the German's team-mate Lewis Hamilton runner-up and Ferrari's Sebastian Vettel third.
Reliability issues with the Renault engine blighted Ricciardo and Red Bull last year but Sunday's performance left the 26-year-old flashing his usual toothy grin.
"I'm really happy, towards the end we were sitting in the top three and I had the fairytale in my mind of being on the podium here but I'm happy," he told Sky Sports.
"Fourth is great but I'm probably more happy with the pace we had.
"Not far off Mercedes and Ferrari and we finished a fair step ahead of Williams and the rest of our competitors.
"To come out here in the first race and exceed expectations, you've got to be happy with that."
It was a forgettable day for his team-mate Daniil Kvyat, however, with the Russian unable to race after his car came to a halt with a power unit problem at the end of the formation lap.
Team principal Christian Horner was nonetheless encouraged by Red Bull's pace, while Ricciardo said they had definitely found "a little bit more" from the power unit.
Happy with that! 4th at home. Good start to the season. Thank you for the home crowd support https://t.co/PPYCS12oXD— Daniel Ricciardo (@danielricciardo) March 20, 2016
He said the team were expecting an upgrade to the Renault package by the Canadian Grand Prix in June, which could offer a further lift in performance.
"And we know there's a good step to come in a few races," he added.
"We can definitely aim for more top fives before the (engine) update ... We're hoping it's going to come sooner rather than later but perhaps Montreal is where we expect it, and from that point, yeah, I want to spray some champagne."