Alexis scored in the 89th minute as Valencia edged out Tenerife 1-0 and condemned their opponents to the drop.
The victory was no less than Valencia deserved in a match they dominated, but Tenerife looked like they would hold out for a point until the very end.
The draw would not have been good enough for Tenerife to avoid the drop, as Malaga took a surprise point of title challengers Real Madrid. That would have left the two clubs level, but they have an equal head-to-head record with Malaga enjoying a vastly superior goal difference.
Roman Martinez tested Miguel Moya from close range in only the second minute of the match, but that would hardly set the tone for a match of few chances.
It took until the 18th minute for Valencia to get a sight on goal, but Vicente saw his effort blocked by Sergio Aragoneses.
Alejandro Dominguez had another sight of goal just before the half-hour mark, but dragged his shot wide from the edge of the area.
But Valencia were enjoying their best spell in the match and moments later, Nikola Zigic had a good chance but failed to trouble the goalkeeper with a close-range header.
Moments later, Vicente whipped in a corner that Ruben Baraja met but he could not find the target from a good position.
Dominguez then sent a free-kick over the crossbar after Hens Juanlu brought down Vicente to earn himself a yellow card.
Just as they had in the first half, Tenerife had an opportunity early in the second, but Alejandro Alfaro's shot from the edge of the area was held by Moya.
At the other end, Jordi Alba's cross found Zigic, but he lashed his shot over the crossbar from close range.
Valencia were piling on the pressure, with Dominguez's shot from the centre of the penalty forcing a fine save from Aragoneses before Vicente sent a follow-up effort over the crossbar moments later.
It was one-way traffic as the hosts continued to throw everything they had at Tenerife.
Pablo Hernandez brought another save out of Aragoneses before Vicente fired a shot wide from the edge of the area.
Tenerife had created nothing the entire half, and when their first chance did come in the 72nd minute, it was a weak shot from Ricardo Leon which dribbled wide.
Back at the other end, Vicente's rising shot was tipped over by Aragoneses as Valencia's frustrations continued.
Zigic then had a chance with a shot from the edge of the area, but he sent it over.
Just when it looked as though Valencia would get no reward for their endeavour, they finally made the breakthrough a minute from time.
Dominguez whipped in an inswinging corner from the left, finding Alexis at the far post and he headed home to snatch the points.