Sheffield, playing the role of enthusiastic spoilers, had dragged themselves level and threatened to win the game. But Goodwin took a poor pass, lurched off balance past a couple of tacklers and claimed the single point that could be of incalculable value when the play-off places are decided.
London could hardly have had a better start, their forwards and particularly Peter Gill and Matt Dunford creating havoc in the Sheffield defence.
It was Gill's long break that set up the position for the first try after five minutes, the Broncos running the ball on the last tackle for Goodwin to send Dunford over for a touchdown improved by Chris Ryan.
Dunford and Gill then set up a second try, scored by Ryan off Gill's clever inside pass, and Timu completed a devastating first 20 minutes by taking Glen Air's reverse pass and crashing through some weak tackling.
The problem was that London then sat back to admire their handiwork. Paul Carr, who is returning to Australia at the end of the season, showed how his nose for the close-range try will be missed by burrowing his way over for two virtually identical efforts.
It looked as though the theory of Eagle's coach, John Kear, that playing in the more intimate surroundings of Bramall Lane would lift them was correct.
Sheffield drew level early in the second half, Dave Watson's kick not looking particularly dangerous until Bright Sodje flashed down the wing past a static defence to score, Martin Wood landing his second goal.
A penalty under the sticks for off-side came as welcome relief for the rattled Broncos, Ryan's kick edging them back into the lead, but Wood's penalty made the scores square with 14 minutes to play. Mark Aston had a drop goal disallowed for a faulty play-the-ball before Goodwin's winner. Even after that London had some nervous moments, notably when Wood dropped the ball as he surged over their line.
They managed to hang on to their precarious lead, though, to draw level with Bradford, but it was so nearly the end of their bid for a play-off place.
Sheffield: Watson; Sovatabua, Morganson, Senior, Sodje; Vassilakopoulous, Stephens; Molloy, Lawless, Laughton, Carr, Shaw, Wood. Substitutes used: Taewa, Cardoza, Aston, Doyle.
London: Toshack; Smyth, Timu, Ryan, Goodwin; Tollett, Air; Young, Williams, Dunford, Millard, Retchless, Gill. Substitutes used: Salter, Beazley, Edwards.
Referee: S Cummings (Widnes).
n Halifax ran in 30 points in 24 minutes to secure the third-place berth by overwhelming Hull 44-24 at the Boulevard.Reuse content