The unmarked Montero should have added a second goal with 15 minutes to go when he nodded over a centre by Sigurdsson from six metres.
"This is the first victory in the Premier League for Swansea against Chelsea and I am happy and proud," manager Francesco Guidolin told the BBC.
"We deserved to score other goals. Safety is almost real but I hope to take other points in the next weeks because it's important to finish well."
Mid-table Chelsea, fielding a young side in the absence of John Terry, Diego Costa, Eden Hazard, Willian and Gary Cahill, lost for the first time in the league under interim manager Guus Hiddink who had gone 15 matches without defeat since taking over from the sacked Jose Mourinho.
"Every defeat hurts but everyone knows the situation we are in so there's a positive side to bringing in youngsters and seeing what they can do," said Hiddink.
"We defended poorly and gave Sigurdsson a very free chance. We were attacking, attacking but in the final third we were not sharp enough."
(Writing by Tony Jimenez, editing by Pritha Sarkar)
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