Leeds, playing under head coach Thomas Christiansen for the first time, went ahead through Phillips' smart finish.
Last season's top scorer Chris Wood then headed in for 2-0, only for Bolton forward Gary Madine to nod home.
But Phillips' tap-in quickly restored Leeds' two-goal lead, and they held on despite Adam Le Fondre's penalty.
The former Reading front man smashed his spot-kick down the middle, with Conor Shaughnessy having fouled Madine almost immediately after coming on.
But the headlines belonged to 21-year-old Phillips, who doubled his career tally for Leeds during the first half to put the visitors in control.
New Zealand forward Wood, who scored 30 goals last season as Leeds narrowly missed out on the play-offs, also opened his account with a clinical header in off the underside of the bar.
But there will be some concern over the fitness of defenders Gaetano Berardi and Matthew Pennington, who were both forced off injured prior to half-time.
Bolton, promoted from League One in May, were direct throughout but struggled to create anything clear-cut from open play - and also lost midfielder Josh Vela through injury at the end of the game.
Kalvin Phillips scored twice as Leeds United opened their 2017-18 Championship campaign with victory at Bolton Wanderers.