A break from Saudi Arabian Hussain Jamaan Alhamdah stretched the pack but Farah moved up through the field to sit on his shoulder as the race entered the final kilometre.
Lagat tracked Farah throughout the second half of the race, but was unable to match his victory over this distance in 2007 and had to settle for a second successive silver.
Farah, who covered the final 400m in 52.87 seconds to become the first British world champion over the distance, added: "It's been hard work and a lot of sacrifices and I have to thank so many people who've been behind me.
"But you can't get carried away. It's a long way [to the London 2012 Olympics]. The aim is to stay injury free. Anything can happen in long distance."
Farah 'can't believe' 5,000m gold (UK only)
Idowu jumped a season's best 17.70m in the third round to stretch his lead in the triple jump, but American Taylor responded with a 17.96m effort.
And that proved the winning leap as Idowu, who improved to 17.77m in the fourth round, was unable to jump further in his final two attempts and he had to settle for silver.
"It wasn't my day," said the 2009 world champion. "I felt I had that distance in my legs but it never came.
"I jumped two season's best. It wasn't enough but hopefully I can do it next year [at the London 2012 Olympics] and get gold.
TOM FORDYCE'S BLOG
“For Farah, it was the culmination of a journey that began when PE teacher Alan Watkinson spotted a skinny kid hanging from the crossbar of the football posts at Feltham Community College and called him down for a chat”
America's third-leg runner Darvis Patton fell over after clipping Britain's fourth-leg athlete Harry Aikines-Aryeetey as he came into the changeover zone.
British head coach Charles van Commenee said: "From my perspective it looked like Harry went way too early.
"I'm struggling to understand always why that baton can't get round."
Aikines-Areetey defended himself, saying: "I don't think I went too soon.
"I felt someone hit me and then saw someone tumble over. As to why Marlon [Devonish] couldn't get the baton to me, I've no idea."
GB'S MEDALS IN DAEGU
- 5,000m gold
- 400m hurdles gold
- 10,000m silver
- triple jump silver
- heptathlon silver
- 1500m silver
- 110m hurdles bronze
women's 4x100m relay team
of Tiffany Porter, Anyika Onuora, Laura Turner and Jeanette Kwakye failed to progress from their heat after finishing fifth.
Kwakye told BBC Radio 5 live: "It's important for us as a team to perform on a world stage.
"Unfortunately we didn't get a time to show what we can do next year but it's about getting the practice as a team ahead of 2012."
BBC Radio 5 live summariser Darren Campbell said: "The British team performed well but the first-to-second changeover [between Porter and Onuora] was where it was all lost and they were never in contention after that."
Britain's David Webb
attained the London 2012 A-standard in finishing 15th in the marathon. His time of two hours, 15 minutes, 48 seconds was a season's best and good enough for 15th.
His team-mate Lee Merrien was 22nd in a race won by Kenya's defending champion Abel Kirui in 2:07.38.
Vincent Kipruto, also of Kenya, took second ahead of Ethiopia's Feyisa Lilesa.
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