The Swede became England's first overseas manager and led them to the quarter-finals in three successive tournaments during his spell in charge between 2001 and 2006.
Eriksson, who also managed Manchester City, Lazio and Roma, revealed in January that he had been diagnosed with terminal cancer and that he likely had "at best" about a year to live.
He died at home in Björkefors, near Sunne, yesterday morning, surrounded by his family.
Beckham posted footage of a recent meeting with Eriksson on Instagram and wrote: "We laughed, we cried and we knew we were saying goodbye. Sven, thank you for always being the person you have always been - passionate, caring, calm and a true gentleman."
"I will be for ever grateful for you making me your captain but I will for ever hold these last memories of this day with you and your family."
The former Liverpool and England striker Michael Owen, who scored a hat-trick in England's 5-1 win over Germany at the Olympiastadion in Munich in 2001 that marked arguably Eriksson's greatest day as a manager, referred to "one of the very best and a man who will be sadly missed by everyone in the world of football".
Kasper Schmeichel, who Eriksson signed for Leicester in 2011, was among others to pay tribute. "Sven was the first manager to truly believe in me and give me a chance at the highest level," the goalkeeper said in a message on Instagram. "He was the first manager to call me worldclass and he made me feel 10 feet tall.
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