Sterling, one of a number of Liverpool players to enhance their reputation last season, gave the hosts the lead midway through the first half after a fine pass from England team mate Jordan Henderson.
A thunderous shot from Nathaniel Clyne drew the visitors level shortly after the break, before Sturridge scored the winner with just over 10 minutes remaining.
Suarez's 31 goals helped Liverpool to second place in the Premier League last season, but the controversial Uruguayan was sold to Barcelona after receiving a four-month ban for biting Italy's Giorgio Chiellini during the World Cup.
His sale helped fund around 100 million pounds ($166.95 million) of new arrivals, but Brendan Rodgers' side largely failed to hit the thrilling heights of last term in their first league outing.
Southampton, themselves embarking on a new period under the guidance of Dutchman Ronald Koeman and after the departures of a number of key names, gave a decent account of themselves in what is expected to be a season of rebuilding.
"They made us work very hard for that, first games are always difficult," Sturridge told Sky Sports.
"The pace of the game was high, we worked hard, we probed a while and when they got the goal it was difficult for us. In the end we ground out the result and that's what matters most."
Three players traded the south coast for Merseyside in the off-season but only Dejan Lovren started, with striker Rickie Lambert on the bench and midfielder Adam Lallana injured.
Clear chances were in short supply for both sides until Henderson won the ball inside his own half on 23 minutes.
He split the Southampton defence with an incisive pass, allowing 19-year-old Sterling to calmly shoot past Southampton's new keeper Fraser Forster into the bottom corner.
Liverpool earned plenty of plaudits for the way they tore teams apart last season, but after going a goal up they allowed Southampton to get a foothold in the match.
James Ward-Prowse sent in some dangerous balls from out wide, but shots on goal were at a premium until Morgan Schneiderlin, who had publicly expressed his desire to join the Southampton exodus, forced a good save from Simon Mignolet shortly before the break.
The Belgian could do little 10 minutes after restart, however, Clyne blasting in after a neat one-two with newcomer Dusan Tadic, the Serb playing a delightful backheel inside the box to set up the right back.
They nearly took the lead shortly afterwards, Tadic and Ward-Prowse combining again to tee up Steven Davis in space, the Northern Irishman firing a tame side-foot shot at Mignolet.
They were made to pay for the wastefulness when Sturridge, who scored 21 league goals last term in partnership with Suarez, flicked the ball home after the impressive Sterling met Henderson's cross.
Schneiderlin nearly snatched a late point, Mignolet tipping his shot onto the underside of the crossbar, with Shane Long sending the follow-up just wide.
Newcastle host champions Manchester City in Sunday's second match.