On the stroke of half-time, Gloucester centre Fuimaono-Sapolu went over for a third Gloucester try to make the score 19-11.
But any hopes the Cherry and Whites had of cruising to victory in the second half were dashed as they repeatedly spurned chances to increase their advantage.
Lewis-Pratt cut the lead with a penalty and then struck a post with another. Centre Burrell was the first to react to the ricochet, collecting the ball and thundering over.
The ensuing conversion put Leeds ahead for the first time with 13 minutes remaining.
But as they tried to run down the clock, the visitors were penalised for an offence in the ruck. Robinson duly stepped up to give Gloucester victory by a single point.
Gloucester head coach Bryan Redpath told BBC Radio Gloucestershire: "The manner in how we won was disappointing. After the first 40 minutes there were too many handling errors again. It's a team game and there are too many individuals at the moment.
"I'm chuffed that we got a win but our ball retention, patience and game management is very poor.
"Mike Tindall [captain] has come into the changing rooms and said what he needed to say which was important from my point of view, because when we stick to basics we create problems."
Leeds head coach Neil Back told BBC Radio Leeds: "The players were unhappy with their first-half performance. I think they showed a lot of character with a spirited second-half performance.
"They played much better rugby, they were more comfortable with the ball in hand and they were much more physical, and that is what you need to be in the Premiership.
"And on the back of that we put ourselves in a position to win this game ... we were two or three minutes away from doing that."
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