Sigurdsson curled over the wall from about 20 yards to cancel out Wayne Rooney’s penalty and move the Swans to within two points of 17th-placed Hull.
Boro thought they had secured a valuable win when Calum Chambers poked in from close range on 77 minutes, only for Jesus to rise and convert Sergio Aguero’s cross with five minutes remaining.
Pedro’s 25-yard stunner, Gary Cahill’s close-range finish and Willian’s tap-in kept Antonio Conte’s side four points clear of second-placed Tottenham, who beat Arsenal 2-0 later on Sunday.
Chelsea’s 3-0 victory at Everton put the pressure on Spurs – but they responded in style with a ninth successive league win to continue their best sequence since October 1960, when they won 13 games in a row.
Ashley Barnes and Andre Gray scored the goals to leave Palace seven points above the bottom three, with four games remaining.