Senegal coach Bruno Metsu hailed his side's qualification to the second phase as a "historic day".
The Dakar Lions stormed three goals ahead through a disputed Khaliou Fadiga penalty and two from Pape Bouba.
But they were clinging on desperately by the end after letting their lead slip.
"It is our first time at the World Cup
and to be part of the 16 best teams in the world is an extraordinary success."
Metsu acknowledged Uruguay's dramatic comeback had made the game one of the matches of the tournament so far.
"It was engrossing, moving, difficult and it was a
fantastic spectacle," he added.
"The goal they got just after half-time really hurt us and it
turned into a heart-stopping match.
"But the players showed they are capable of doing great
Bouba Diop himself said felt he and his team-mates deserved to go through.
"It's a great feeling for us and a great achievement," he said.
tough for us, but we gave it
Fans in Dakar have poured out onto the streets in celebration.
"Nobody's working, it's a holiday, the streets are full, sound systems are at full blast," one fan told BBC Sport Online.
"Even the French people who live in Dakar are wearing red, gold and green.
"We're unbeatable at the moment."