Aiden O'Brien opened the scoring in the first half after an assist from Jed Wallace who twice came close to scoring himself.
Both sides had plenty of chances in an end-to-end game before Preston earned a crucial point through Robinson 10 minutes from time to maintain his side's push for a top-six spot in the Sky Bet Championship.
Early on Preston produced a dangerous attack when Billy Bodin made a break down the right wing but his cross narrowly missed an unmarked Jordan Hugill inside the six-yard box.
Millwall, playing their 2,000th home game in the English Football League, had a flurry of chances in the first 20 minutes, most notably when James Meredith's long throw-in found Steve Morison on the edge of the box but his header went straight into the hands of goalkeeper Chris Maxwell.
Moments later, the visitors intercepted the ball in their own half and Paul Gallagher broke away through the middle before offloading to a determined Bodin, whose powerful shot went wide from a deflection.
Preston had another chance on the counter midway through the first half, this time Robinson made an assault down the left-hand side but his shot flew over the crossbar.
The hosts earned themselves a free-kick before the 40-minute mark after Morison clashed with Tom Clarke when the pair contended for a high ball but the set-piece was wasted. However, the Lions took the lead just before half-time when an off-balance O'Brien connected with Wallace's low cross and slotted the ball into the net from close range.
Millwall maintained the momentum in the second half as Wallace put the ball past two Preston defenders and forced goalkeeper Maxwell to meet him outside the box for a clearance.
Five minutes later, George Saville nearly doubled the hosts' lead when his shot from outside the box missed the target by inches as it whizzed past the right post.
An equaliser came for the visitors on 80 minutes when Robinson connected with a sharp pass from Greg Cunningham and chested the ball straight past the goalkeeper.
Both teams hunted for the winner as substitution Daniel Johnson broke away twice for Preston but a sound Millwall defence were able to hang on to the draw.