The £5m signing put the Swans ahead when he left new Reds defender Mamadou Sakho flat-footed to calmly stroke a left-foot effort past goalkeeper Simon Mignolet.
Liverpool had not previously conceded in the Premier League and they were gifted a way back into the game little over 100 seconds later by Shelvey.
His careless pass back towards goalkeeper Michel Vorm was seized upon by Sturridge, who netted his fourth top-flight goal of the season with a clinical finish.
The Reds dominated the opening half hour as the vibrant trio of Moses, Philippe Coutinho and Sturridge pulled apart the Swansea defence with pace and precision. And they should have doubled their advantage when Moses and Sturridge combined.
On-loan Chelsea winger Moses
surged past two defenders near the left corner flag and dinked in a perfectly weighted cross towards an unmarked Sturridge.
The England striker seemed certain to score but his header was directed too close to Vorm, who made a magnificent save.
We showed 'character' - Brendan Rodgers
Shortly before half-time, they did take a deserved lead. Shelvey sloppily handed possession to Moses, who drove at the backpedalling Swansea defence before accurately shooting into the bottom corner from the edge of the penalty area.
Victory would have given Liverpool their best start to a season since 1990-91 and for the opening hour they seemed firmly on course to achieve that.
But Coutinho's departure early in the second half with a shoulder injury following a robust challenge from Swans skipper Ashley Williams disrupted their fluency.
Swansea, playing the first of seven games in 21 days, had not created any clear-cut second-half chances before Michu's leveller, but they had been encouraged to press forward by Liverpool's willingness to sit back on their lead.
They were rewarded when Shelvey flicked a right-wing cross into the path of Michu, who swept a first-time shot beyond Mignolet.
Jonjo Shelvey will learn - Michael Laudrup
The leveller switched the momentum of the match firmly in Swansea's favour and, as the encounter moved into the closing stages, Michael Laudrup's men looked the likeliest to grab a dramatic winner.
Shelvey sent a low free-kick fizzing into the arms of Mignolet, before substitute Jonathan de Guzman twice went close.
Swansea manager Michael Laudrup:
"We had to perform better in the second half and the boys did a good job.
"We got the equaliser, we could have won the game but I'm pleased with the result after all."
Liverpool manager Brendan Rodgers:
"I thought for the first 65 minutes we were very good. We showed incredible character to get back level so soon after they scored.
"I was disappointed with their second goal and then the momentum changed for the last 25 minutes and we had to show character and resilience.
"But we showed tonight our game is getting better and we have the character to get a lot of results."
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