Kenyan Mary Keitany retained the women’s title she won in record time last year.
Six-time Paralympic gold medallist David Weir claimed a record-equalling sixth men’s wheelchair crown.
And fellow Briton Shelly Woods won her seventh wheelchair title after her main rival crashed close to the finish.
Farah, 32, was made to work hard as Biwott, runner-up at the 2014 London Marathon, led for much of the race.
But the Olympic and world 5,000m and 10,000m champion produced a trademark sprint finish to take the lead in the final 400 metres and win in 59 minutes 22 seconds – the fastest half marathon by a British athlete.
“The crowd were amazing,” said Farah. “If it wasn’t for them edging me towards the end it definitely would have been a different result.
“I want to come back and see if I can do a hat-trick.”
Keitany enjoyed a more comfortable morning as she won in 1:07.32, nearly three-and-a-half minutes ahead of British runner-up Gemma Steel.
The 2012 London Marathon winner, who beat Paula Radcliffe’s previous record when she won last year , said: “It’s a really amazing day for me. I love the city and the people here.”
Briton Alyson Dixon, 36, finished fourth, behind two-time New York marathon winner Jelena Prokopcuka of Latvia.
Weir dominated the men’s wheelchair race from the start, finishing in 42.46, one minute and 58 seconds ahead of fellow Briton Simon Lawson.
The 36-year-old said: “I had a bit of a problem in the first three or four miles with stomach cramps.
“I got a second wind after that but it cost me a minute – otherwise I could have broken the record. But was it was a fun day.”
There was drama in the women’s wheelchair race when leader Margriet van den Broek hit a hay bale near the end.
Woods capitalised to win in 53.38, just ahead of course record holder and 2009 winner Amanda McGrory of the United States, with Van den Broek of the Netherlands finishing third.