Dickinson dispatched his spot-kick into the bottom-right corner in the 27th minute with Town goalkeeper Ben Smith diving in the opposite direction, after Jermaine Grandison had needlessly shoved Harris in the box.
Phillips stabbed home a loose ball after Smith fumbled Ryan Hall's free-kick in the first minute of stoppage time before Harris put the final nail in the coffin with a last-gasp penalty following Shane Cansdell-Sherriff's foul.
In Friday night's other game, Swindon boss Paolo di Canio was left licking his wounds after gritty Macclesfield produced a powerful second-half display in a 2-0 victory.
Macc took the lead in the 67th minute when on-loan striker George Donnelly flicked on a long ball from youngster Elliott Hewitt honing in on goal from 40 yards, before drilling a low drive through the hands of substitute goalkeeper Mattia Lanzano.
The home side then sealed the game in the 83rd minute with a bullet drive from Ross Draper.
Di Canio was left cursing referee Steve Rushton when Swindon midfield playmaker Matt Ritchie danced past keeper Jose Veiga only to have his heels clipped. The youngster lost his momentum but the referee waved away appeals for a penalty as he stayed on his feet.