Solano stroked the spot-kick to Robinson's right as the keeper dived the other way.
Newcastle put their midweek Cup disappointment behind them with their second Premiership win over Leeds in the space of 25 days.
Goals from Nol Solano, Clarence Acuna and Shola Ameobi left Leeds marooned in the bottom third of the Premiership despite Robbie Keane's second-minute strike.
After Solano had equalised on three minutes, Acuna scored just before half-time before Ameobi wrapped things up at the death.
Leeds put Newcastle under constant pressure in the second half, winning an incredible 16 corners as the visitors sat back, content to rely on Gallacher and Ameobi's strong running on the break.
But when the latter lashed Solano's knock-down home from the edge of the box on 86 minutes, it sealed a thoroughly deserved win for Bobby Robson's injury-ravaged side.
The game had begun at a frantic pace.
With a minute on the clock, Viduka squeezed past the ponderous Goma on the left and pulled the ball across the face of goal.
Keane mis-kicked, Bakke did the same and Keane poked the loose ball high into the net from six yards.
With the roar of the home fans still in their throats, Gallacher stumbled into space in the Leeds box and saw his attempted cross blocked by the hand of Danny Mills. Penalty.
Solano stroked the spot-kick to Robinson's right as the keeper dived the other way.
Sprawling save
Viduka, Leeds' top scorer this season by a distance, turned splendidly in the box only to see Shay Given, in for the injured Steve Harper, spring to his right to deflect the ball away.
The Irishman produced an even better stop on 27 minutes.
Then a mistake by Woodgate let in Gallacher, only for Robinson to save the defender's blushes with a sprawling parry.
Perfectly
Acuna struck with the home defence clearly thinking about their half-time cuppas.
Speed popped a delicate ball through and Acuna, falling under Dacourt's challenge, managed to dink his shot in a perfectly over Robinson and gently into the net.
Leeds:
Robinson, Mills, Woodgate, Radebe, Matteo, Bowyer, Dacourt, Bakke, Wilcox, Keane, Viduka
Subs: Martyn, Harte, Smith, Batty, Burns.
Newcastle:
Given, Barton, Hughes, Goma, Quinn, Solano, Lee,
Acuna, Speed, Ameobi, Gallacher.
Subs: Marcelino, Bassedas,
Cordone, Lua-Lua, Karelse.
Referee: A Wiley (Burntwood)