Boro were successful with all of their penalties, having come from behind twice in normal time with goals from Ashley Fletcher and Marcus Tavernier cancelling out efforts from Robinson and Tom Barkhuizen.
Middlesbrough manager Tony Pulis changed all 11 of his players from the weekend’s goalless draw with Swansea, and they would have been ahead after just six minutes had Paddy McNair not produced a tame side-footed effort when in space in the area.
Dimi Konstantopoulos was called into action at the other end shortly after, keeping out Paul Huntington’s header from a corner, and with Middlesbrough struggling to come to terms with their switch to a five-man defence, Preston enjoyed plenty of action down their left.
Huntington glanced Ryan Ledson’s cross narrowly wide of the target, and Pulis was forced to switch his full-backs before the break in an attempt to tighten things up at the back.
Preston broke the deadlock shortly before the half-hour mark. Alan Browne squared the ball for Robinson, who rolled around teenager Nathan Wood before firing into the corner.
Boro were level within seven minutes, Lewis Wing releasing Fletcher through the middle, and the striker calmly rounded Chris Maxwell before slotting home.
Rudy Gestede flashed a header wide on the stroke of half-time and Barkhuizen cracked a 49th-minute drive against the base of the left-hand post.
Moments later, and it was Boro hitting the woodwork through Gestede.
Preston reclaimed the lead shortly after the hour mark. Konstantopoulos dived over the ball, enabling Barkhuizen to find the net from 25 yards.
Having gone ahead for a second time, Preston were pegged back once more just three minutes later.
Tavernier picked up the ball in space, and found the top right-hand corner from outside the box.
Preston North End: Maxwell, Clarke, Storey, Huntington, Hughes, Pearson, Browne (Harrop 63), Barkhuizen (Moult 85), Ledson, Barker (Burke 81), Robinson. Unused substitutes: Rudd, Earl, Johnson, Gallagher.
Middlesbrough: Konstantopoulos, Batth, McNair, Wood-Gordon (Chapman 72), Fry, Tavernier, Leadbitter, McQueen, Wing, Fletcher, Gestede. Unused substitutes: Malley, Liddle, Lonergan, Spence, Walker, Stubbs.
Referee: Oliver Langford (W Midlands).