Diego Costa and Manucho scored either side of half time with a header and sweeping finish after a counter-attack to reward a spirited display by the hosts.
A stunning long-distance effort from Juan Cala gave Sevilla hope seven minutes from time and it took desperate saves from Valladolid goalkeeper Jacobo to deny Jesus Navas and Luis Fabiano in a nail-biting finish.
Sevilla remained fourth on 51 points, in the last Champions League qualification place, but could drop back if fifth-placed Real Mallorca get at least a draw at Real Zaragoza on Wednesday.
Leaders Barcelona, with 80 points, host Deportivo La Coruna on Wednesday and would open a six-point lead over Real Madrid with victory. Their title rivals do not play in Almeria until Thursday.
Valladolid called in former Spain boss Javier Clemente as their third coach of the campaign last week to try and save them from relegation, and after a draw at the weekend he earned his first victory at the helm.
Clemente's side stayed 19th in the 20-team league with 28 points after 18th-placed Tenerife earned their first away win of the campaign, 2-0 at Sporting Gijon.
Alejandro Alfaro and Roman Martinez netted late goals at the Molinon to move the islanders on to 29 points, two short of 17th-placed Malaga and safety. Malaga visit Osasuna on Wednesday.
In the late game, Villarreal's Spain goalkeeper Diego Lopez had an unfortunate night as his side lost 3-0 in a pulsating encounter in Getafe's rainswept Coliseum.
First he fumbled a long-range shot to gift Nicolas Fedor 'Miku' the opener, and then he could only push Mane's freekick against the post, the rebound going in off his back.
There was nothing he could do about the third, however, as Fedor scored his second of the night on a two-against-one breakaway near the end.
Getafe climbed to seventh with 47 points one ahead of Villarreal in eighth.
