Nigeria, the 1996 Olympic football champions, are now the first African country to complete an Olympic medal sweep having won a sliver medal in Beijing in 2008.
Aminu Umar had hit both posts before Sadiq Umar scored from close range in the 34th minute after captain Mikel John Obi's fine run.
Four minutes after the break, Mikel's strike was half-punched away by Lopez and it fell for Aminu Umar, the Turkey-based forward, to calmly finish inside the box.
Sadiq Umar got his second and his fourth of the tournament in the 56th minute to set the Eagles in flight and put them 3-0 up.
But Anthony Lozano pulled a goal back after 70 minutes, and Honduras' late fight-back was firmly on when Marcelo Pereira scored with four minutes left on the clock.
The Dream Team VI held off their opponents' late rally to win the country's first medal of the Rio games.
Victory gave Nigeria - beaten by Germany in the semi-finals - their place in history, by winning all three men's football Olympic medals.
At Atlanta in 1996, they became the first African team to win the gold medal.
They also finished runners-up at the 2008 Beijing Olympics to take silver, where current under-23 coach Samson Siasia also coached the 'Dream Team' in China.
Sadiq Umar scored a brace as Nigeria won an Olympic bronze medal in the men's football with a 3-2 win over Honduras in Belo Horizonte.