John McGreal should have put the Clarets in front after 35 minutes as Garreth O'Connor's free-kick found its way to the unmarked Burnley defender who shot wide of the target.
The home side scored the only goal of the game six minutes before half-time with Spicer sending a glancing header from a Michael Duff cross past Leicester keeper Rab Douglas.
Douglas was beaten again three minutes later by former Foxes striker Ade Akinbiyi but referee Matt Messias disallowed the strike for a harsh infringement by the Clarets' top scorer.
Akinbiyi continued to torment the Leicester defence and twice had opportunities to make the game safe for the home team.
Firstly he was denied by Douglas and then, when he did succeed in getting the ball beyond the Leicester keeper, he was prevented from scoring the goal which his performance deserved by the post.
Leicester enjoyed plenty of possession and fought hard for an equaliser with Burnley's on-loan goalkeeper Lee Grant denying Leicester strikers Dion Dublin and Mark De Vries.
At the other end, Burnley again had the ball in the back of the net as from a free-kick routine Micah Hyde's mis-cued effort was deflected by McGreal into the path of fellow central defender Keith Lowe only for a flag to deny the on-loan Wolves player.
Leicester almost grabbed the equaliser in stoppage time as Ryan Smith's corner was headed inches wide by Stephen Hughes but the home side held on to earn their fourth successive victory.