Two goals in two minutes early in the first half appeared to have put Leeds in control but a similar burst from the visitors late in the second saw them level at 2-2.
Smith's late goal, however, earned three points that brought the Yorkshire club level with Leicester on 28 points. They stay second-last on goal difference but are now only two points adrift of Portsmouth, who are 17th, the final place of safety.
Leeds were bottom of the league and facing financial ruin two weeks ago but, with a financial takeover safely negotiated and an improvement in their results on the pitch, they can approach the run-in with real belief.
ACROBATIC KICK
Defender Michael Duberry, making his first appearance since January, headed in Seth Johnson's deep cross in the 11th minute and two minutes later Smith found Mark Viduka inside the area and the big Australian striker scored with an acrobatic overhead kick.
Goalkeeper Ian Walker then produced several good saves as Leeds continued to make the running.
However, Leicester, who had won one and drawn five of their last six league games, hit back when poor marking allowed Paul Dickov (77) and Muzzy Izzet (79) to score with well-taken shots.
Leicester could even have won it as Robinson had to make an excellent diving save to deny Peter Canero but seconds later Leeds had regained the lead as Viduka expertly cushioned the ball off his chest and into the path of Smith, who beat Walker on the half-volley.
"It's probably one of the most important goals I'll get and hopefully at the end of the season it will be the three points that will keep us up," Smith told Sky Sports.
The England striker hit the headlines last week when he said he would leave the club he has been with since he was a trainee if they were relegated and accused some of his team mates of "bottling" under pressure.
"I've said a few things that upset a few people but they needed to be said and sometimes the truth hurts," he said.
"But I got the response we wanted tonight. We showed great character to come back and if we can keep that up until the end of the season I think we'll be alright."
The only black mark for Leeds was the injury-time dismissal of Viduka, who was shown a second yellow card for kicking the ball away after a free kick was awarded against him.
