Reeling from a rudderless performance in Sunday’s Premier League loss at West Ham, a much-changed United side showed few signs of improvement in Wednesday’s third-round clash against the side from 20 miles away.
Back-up goalkeeper Romero’s save from Jimmy Keohane’s attempt was the difference as Solskjaer’s side emerged as 5-3 victors, but the outcome could not disguise a poor performance against the side 17th in League One.
United were toothless and underwhelming throughout, with World Cup winner Paul Pogba, back after an ankle injury, wasting the best chance of a one-sided first half.
Greenwood struck in an eerily similar way to last week’s narrow win against Astana, only for impressive 16-year-old Matheson to find the net and delight the 7,000-or-so visiting fans.
Dale dug deep and took the match to penalties, with Romero’s low save on Keohane the only difference.
The visitors were given a hero’s reception at full-time given the way they pushed United all the way.
Solskjaer made nine alterations for the tie and surprisingly named 21-year-old defender Axel Tuanzebe captain against a side with a 16-year-old at right-back.
Matheson’s confident early flick away from Marcos Rojo showed that the teenager was not cowed by the grand setting – nor would the rest of his team-mates given the way they hassled and harried.
A slither of quality denied them creating a clear-cut opportunity in a bright start against an incoherent United side, who peppered the Rochdale goal as the first half wore on.
Many of those chances were from distance and those from closer range lacked a killer instinct, with Robert Sanchez denying a header from Marcos Rojo – a player that had numerous first-half attempts.
Pogba should have put United ahead in the 34th minute only to head over Greenwood’s beautiful clipped cross, with the teenage forward denied by the Rochdale goalkeeper when he burst through.
Jesse Lingard glanced a header just over as the hosts approached half-time on top but scoreless.
Brandon Williams came on for his debut after Phil Jones appeared to take a knock and Andreas Pereira’s deflected effort just went wide.
But every missed chance gave Rochdale increased hope and it took an acrobatic goal-line clearance from Wan-Bissaka to prevent Camps after United academy graduate Ollie Rathbone danced through.
Camps fizzed over moments later and kicked United into a response, with Greenwood denied by Sanchez at his near post and Lingard striking just across the face of goal.
Greenwood stung the palms as the hosts upped the ante, finally breaking the deadlock in the 68th minute. Lingard played out to Greenwood on the right-hand side of the box, with the forward cutting in and firing a low shot home.
But relief would be short-lived as Rochdale stunned Old Trafford with a memorable leveller.
A high cross into the box was met by 16-year-old Matheson, whose effort into the ground beat Romero and rippled the back of the net.
The hosts heaped on the pressure in a bid to steal a late winner, with Greenwood, Wan-Bissaka and Fred among those who failed to convert.
Substitute James and Pereira were denied by Sanchez in stoppage time as the match went to spot-kicks.
The first four attempts all found the net, before Romero got down to deny Keohane with a one-handed stop. It proved the difference as James wrapped things up to send United through.