Saad and Jonathan Toews scored for the Hawks, who evened the best-of-seven series 2-2.
Corey Crawford finished with 24 saves. Game 5 is set for Saturday in Tampa Bay at Amalie Arena, with Game 6 back in Chicago where the Stanley Cup will be present.
Saad came from below the goal line, and picked up the puck after Lightning goaltender Andrei Vasilevskiy poke-checked him, and he knocked it past the rookie net-minder five-hole for the game-winner.
Patrick Kane recorded an assist on Saad's goal for his 112th playoff point, moving him into fourth in franchise history, passing Steve Larmer.
Vasilevskiy, who got the start after Ben Bishop was unable to play because of an undisclosed injury, according to the Lightning, made his first-career postseason start -- on the road no less.
The Hawks made things easy for the rookie early on with a franchise low two shots on net in the first period. The Lightning tightened up their defense and stifled Chicago.
The Hawks coach Joel Quenneville decided to install a major shakeup of his lines and completely mixed up all four. Patrick Sharp skated with Toews and Marian Hossa. Kane was with Saad and Andrew Shaw. Marcus Kruger centered Andrew Desjardins and Teuvo Teravainen while Antoine Vermette was with Brad Richards and Kris Versteeg.
But the Hawks finally broke through in the second period.
After a few near misses from Sharp and Kruger, Toews stuffed in a puck off a pass from Sharp around the net at 6:40. It is the first time the Hawks have scored the first goal in the series. The Lightning are 11-1 this postseason when scoring first, their lone loss came in Game 1 against the Hawks.
Oduya shot from the blue line re-directed off Kruger at 14:32, and rang off the post. Minutes later, Sharp closed in on a breakaway only to have his shot ring off the same post. Toews then knocked in a rebound from Sharp for his career-high 10th goal of the postseason.
The lead wouldn't last long for the Hawks.
Valteri Flippula wrapped around the net, followed by Desjardins and Brent Seabrook, and sent a pass between the two Hawks to Alex Killorn for a wide-open net with Crawford expecting the wrap-around. Killorn sent the puck past Crawford to tie it 1-1 at 11:47 in the second.
Then Saad would score to give the Hawks the lead in the third. Saad had given the Hawks a 2-1 lead in Game 3, only to have the Lightning respond 13 seconds later.
Since 2009, the Hawks are 9-0 in the playoffs when a series is tied 2-2, and are 18-1 in those games.