The 29-year-old, who won the Valencia marathon last December, came home alone in blazing sunshine ahead of Uganda's world half-marathon record holder Jacob Kiplimo.
Defending champion Alexander Mutiso Munyao edged Abdi Nageeye for third in a photo finish, while four-time champion Eliud Kipchoge finished sixth.
"I am very happy to have won a major marathon," said Sawe, who won the Valencia marathon last December.
"I was well prepared for this, that is why I was so relaxed and confident.
"Now this gives me hope that further success in marathons will come."
At the halfway mark, the lead group, which included Kipchoge, Olympic champion Tamirat Tola and Kiplimo, timed outside world record pace at 1:01:30.
Kipchoge, though, was dropped as the runners approached the 90-minute mark.
It was at the drinks station shortly after that that Sawe made the decisive break. Unlike the others, he did not take a drink and stole a march on them.
Kiplimo was caught unawares as he was at the back of the group, but the 24-year-old battled through to suggest his day will come.
Sawe, though, never had a concern from that moment on, looking fresh as a daisy as he broached the winner's tape on the Mall in the shadow of Buckingham Palace.