Precious little happened in a goalless first half at Anfield but Swansea, who began the day bottom, silenced the Kop when Spanish forward Fernando Llorente toe-poked home two minutes after halftime and headed in another five minutes later.
Liverpool retaliated, two goals by Roberto Firmino dragging them back, the first a header from James Milner's 55th-minute cross and the second a sumptuous half volley.
Klopp's side were the first top-flight side to score 50 goals this season but their defensive deficiencies again proved a thorn in their side as memories of a 4-3 defeat at Bournemouth in December came flooding back minutes later.
A Liverpool victory had looked the most likely outcome with half an hour left but Swansea had other ideas and from a rare foray forward the ball fell to Sigurdsson and the Icelander put the Welsh side back ahead.
Liverpool could not respond, leaving Klopp crestfallen.
"The most disappointing moment was the third goal and I can't explain it as we had so many chances to challenge," the German said.
"It's really difficult to accept at this moment. It is fair Swansea won, no - but was it deserved, yes."
Improving West Ham United won for the fifth time in seven league games to move into the top half - Andy Carroll again on target with two goals in a 3-1 victory at struggling Middlesbrough.
Sunderland sunk to the bottom after a 2-0 defeat at West Bromwich Albion for whom Darren Fletcher and Chris Brunt were on target.
Bournemouth drew 2-2 with Watford and Crystal Palace ended the day in the bottom three for the first time this season after a tame 1-0 home defeat by Everton, Seamus Coleman thumping home a late winner.