Spurs won 1-0 courtesy of Peter Crouch's effort to clinch fourth place.
"The supporters have been waiting for this for a long time, as the players have," King told Sky Sports 2. "We've worked hard all season for this and you can see how much it meant today."
King played down the Champions League place as Tottenham, even if they overhaul north London rivals Arsenal and finish third, will have to go through qualifying to reach the group stages of Europe's elite competition.
"At the moment we're in fourth place. We still have to qualify for the Champions League," added King, who plays on despite a chronic knee injury.
"But that's why I still work hard to get on the pitch, because of moments like this."
King added: "It's been a topsy-turvy campaign this season, but I think we've shown the most consistency."
"It's possible, they have to lose and we still have to beat Burnley," said King.