The Lightning jumped out to a fast start, with Connolly and Ondrej Palat scoring in the first 6:51 of the game.
Victor Hedman assisted on each goal for Connolly, who weaved through two defenders and pushed a wrist shot past goaltender Carter Hutton for his second score at 18:41 of the middle period.
Stamkos' first goal of the preseason lifted Tampa Bay to a 3-0 lead early in the second.
Matt Cullen scored the Predators' only goal in the third period against backup goaltender Cedric Desjardins.
Riku Helenius made his first start in goal for the Lightning and stopped all 15 shots he faced in a little more than 30 minutes of ice time. Hutton gave up four goals on 17 shots in 40 minutes before being replaced by Marek Mazanec.