Three of the goals were scored in the first half as Eddie Howe's men surged up to fifth in the early season table, with a double from Fraser followed up by Joshua King's penalty.
Leicester never looked like getting back into the match after the break and any chance of a comeback was seemingly ended when captain Wes Morgan was sent off with 21 minutes remaining.
Adam Smith struck nine minutes from time as Bournemouth made the most of their numerical advantage and, although the Foxes pulled goals back through James Maddison and Marc Albrighton, Claude Puel's side has now lost three of its five top-flight games this season.
Bournemouth netted first in the 19th minute through Fraser who, having earlier had a penalty claim turned down after a challenge from Ricardo Pereira, took a pass from King, cut inside from the left and fired a low, right-foot shot into the bottom corner from 18 yards.
Seven minutes before the break, Bournemouth doubled its lead when Callum Wilson's throughball caught out the visitors' high defensive line and that man Fraser slipped a left-footed finish through the legs of Kasper Schmeichel.
It was three for Bournemouth in the 41st minute, Pereira penalised for handball as he battled for possession in the area with King, who stepped up to take the resulting spot-kick himself and calmly sent Schmeichel the wrong way with a low effort.
Leicester brought on Kelechi Iheanacho and Albrighton around the hour mark, but a miserable day at the office got worse when Morgan saw red, the defender getting a second booking for what referee Craig Pawson deemed was a foul on Smith as he put in a series of challenges around midfield.
Smith then had another key involvement with nine minutes to go, firing a first-time finish past Schmeichel from 15 yards after an excellent assist from Fraser.
Leicester, however, ensured some respectability in the scoreline with a pair of strikes in the final two minutes. Maddison sent Begovic the wrong way from the spot after Diego Rico had fouled Pereira, before Albrighton's glancing header from Iheanacho's cross found the bottom corner.