Llorente got the better of Phil Jagielka to nod Jordan Ayew’s cross past goalkeeper Maarten Stekelenburg from close range – the Spaniard’s 13th league goal of the season.
Mason Holgate’s last-ditch challenge prevented Alfie Mawson from making it 2-0 before Ayew hit the post from 16 yards.
Hull, who earlier lost 2-0 at home to relegated Sunderland, drop into the bottom three, while this result also means Middlesbrough will be relegated to the Championship if they lose to leaders Chelsea at Stamford Bridge on Monday (20:00 BST).
Everton were poor throughout, Romelu Lukaku going closest when he blazed into the side-netting after a powerful run.
At the other end, Stekelenburg produced a fine stop to deny substitute Leroy Fer from making it 2-0 for Swansea, who are unbeaten in three league games.
Swansea’s fate in their own hands
On a dramatic day at the bottom of the table, Swansea took advantage of earlier slips by Hull and Crystal Palace to leave their Premier League future in their own hands.
The Swans are away to relegated Sunderland on 13 May and at home to eighth-placed West Brom on 21 May, the final day. If they win both games they will stay up.
Hull are away at Crystal Palace, who are four points above the relegation zone after a 5-0 defeat at Manchester City, on 14 May before hosting second-placed Tottenham on the last day.
Palace’s final match of the season is at Manchester United, who are unbeaten in their past 24 league games.
