Manipur-based club Neroca FC had finished at the bottom of the table in the last I-League.
Fulham joined Sheffield United and West Bromwich Albion in going down to the Championship, wrapping up the Premier League's relegation battle.
Burnley were sharper and more committed but mid-table Wolves contributed to their downfall with some sloppy defensive work at an empty Molineux.
Lambert said the club can still avoid the drop despite being five points from safety with only four Premier League games left to play.
He took over at Swansea in January, replacing Bob Bradley, with the club bottom of the Premier League and tipped as odds-on favourites for relegation.
American owner Ellis Short doesn't mask the problems engulfing the club, saying "significant work" is now required to secure an instant return to the Premier League.