Legault opened the scoring on a two-man advantage at 13:11 of the first period, redirecting a Dylan Wruck shot just inside the left post behind Bronco netminder Eetu Laurikainen.
Legault struck again on the power play at 3:14 of the second period, this time slamming home a rebound off Laurikainen's left pad to make it 2-0 Oil Kings.
Zac Mackay scored for the Broncos at 17:35 of the second period, tracking down his own rebound and catching Edmonton goalie Laurent Brossoit off his post, and banking it off his left pad.
Edmonton (14-5-4) killed off all five Swift Current power plays, giving them 37 straight kills and eight consecutive games without allowing a power play goal.
Laurikainen appeared in his 16th straight game for the Broncos (9-12-5) and stopped 42 shots.
Brossoit made 26 saves for Edmonton.