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Wing Leon Lloyd went over in the corner early in the second half as Leicester took charge of a nervy game.
Biggs replied with a solo effort from halfway which proved the difference.
Leeds coach Phil Davies sent out a warning to the Tykes' relegation rivals after his side's victory, insisting they would not give up the fight.
"I think we deserved to the win, we created a lot of chances," Davies said. "It was a great victory for the club but we mustn't get carried away.
"We are still four points behind but now we have got the other teams looking over their shoulders.
"This was a must-win game but now we've got to carry it on."
Leicester Tigers boss Wells said his team had been outfought by Leeds.
"Leeds tackled everything and made it difficult for us. We couldn't play the rugby we wanted to, so credit to them for making us play the way we did.
"When things go well for us we look a good side but when we have to dig in we need to show more composure.
"Perhaps it would have been better if we hadn't had the players back from international rugby," he said.
