Reid, who still has a year of his contract to run, needed a win to keep any hope of survival alive, but a brace from Rickie Lambert and a goal for Ryan Dickson put the Saints in control. Yannick Bolasie grabbed a consolation in stoppage time but it was not enough to prevent an inevitable drop for the Pilgrims.
Reid said: "We gave it our best shot - I couldn't have asked for any more from the players. Obviously the players are devestated, nobody likes to go down and they have given it their best."
He added: "They have no need to reproach themselves. What's happened this season has been really, really difficult for a real good set of players and I have told them that.
"They should be proud, they have done magnificently when you consider they have not been paid since January.
"They have struggled through and given me, as a manager, everything they have got. It wasn't good enough to keep us up but it wasn't for lack of effort.
"It was just one hurdle too far, we just couldn't climb that last hurdle.
"The fans have been magnificent, so has everyone at the club and now I just hope we can go out with a win against Orient at home on Saturday and then see where we go from there."
Southampton will also not be playing in League One next season as the victory all but confirmed their promotion to the Championship.
Huddersfield can still draw level on points with a win on Saturday but they would need a swing of 18 goals to deny Nigel Adkins a promotion.
Source: PA
Source: PA