This demonstrated that Moyes could mastermind a win over significant opposition. Liverpool posed a real threat for sustained periods, midway through the first half, and then during a vibrant second half. Brendan Rodgers had set Moyes a tactical challenge, with Liverpool lining up 3-4-1-2 with José Enrique frequently prominent at left wing-back and Moses roaming through the middle in support of the two strikers, Daniel Sturridge and Suárez.
Suárez deserves particular acclaim. He comes with much baggage, probably hand-luggage next summer, but his commitment was obvious when reunited with that famous Liverpool No?7 shirt. Barring Uruguay duty, Suárez has been in exile, serving a 10-match suspension for biting Chelsea's Branislav Ivanovic, yet his rustiness was only occasionally obvious, and mainly late on when he was running on willpower and adrenalin. Otherwise, his stamina was remarkable. Evans and Smalling could never relax with Suárez around. He almost equalised with an exquisite free-kick.
From the start, Suárez was assailed with all manner of verbal barbs by the home supporters. His name was sung by the 7,200 visitors, many of them up in the top tier of the East Stand where they held up a banner spelling out "We Are The Famous Kopites".
Moyes's players got touch-tight to Suárez at times. Evans clipped his heels. Smalling nipped in ahead of him to reach the ball. United seemed to have settled and Nani and Rooney threatened Simon Mignolet's goal. Gradually, Rodgers's side took control of midfield. Lucas anchored with his usual nous and simple distribution, fetching and carrying for Steven Gerrard, who kept stroking the ball wide. United chased shadows. Suárez wriggled in from the right, looking to tee up Gerrard but Rafael intervened. These were nervous times for United. Enrique enjoyed his more advanced role down the left, lifting in crosses. One ball was cleared by Alex Buttner as Sturridge lurked. Enrique then ran in, passing to Suárez, who tried to usher the ball on to Moses. Giggs, showing all the anticipation acquired over 22 years at the top, sprinted back to intercept.
Still Liverpool attacked. Sturridge stabbed a shot wide. When Evans misjudged a long angled ball from Enrique, Suárez met it with his right foot, his touch slightly heavy, sending the ball running towards De Gea. Suárez still had every right to go for the ball, stretching in and accidentally catching the United keeper.
The sliding Smalling then did well to regain possession from Sturridge, the ball hitting his hand according to the Liverpool striker. Liverpool fans were in good voice, questioning Moyes's managerial credentials. Moyes appeared in the technical area, attempting to inject a greater sense of urgency in his players.
They responded, charging down the right, a Rafael back-heel causing brief anxiety in Liverpool's defence. A few late challenges were going in but the excellent Mark Clattenburg understood the intensity of this fixture and let the game flow. Late in the half, Clattenburg decided he had shown enough leniency and started showing cards. Lucas, having fouled Jones, went into the book followed by Giggs for bringing down Moses.
Clattenburg noted down a goal early in the second half. United emerged from the tunnel far more assertive, taking the lead as Moyes was still walking to the dugout. Rooney took the corner, reading Hernández's clever run. The Mexican escaped far too easily from Enrique and ran towards the incoming ball, greeting Rooney's delivery with a flicked finish.
Whatever the undeniable and expensive lapse of concentration by Enrique, Hernández's goal was rooted in his stealthy movement and ability to twist his body to meet the ball. Moyes punched the air with both fists in celebration.
Liverpool rallied as a good cup-tie broke out, flowing from end to end.
Jordan Henderson placed a shot wide. Shinji Kagawa, suddenly showing his danger through the middle, advanced and let fly, the ball clipping Mignolet's bar.
Enrique flashed a shot wide, angering Gerrard who wanted the move to continue. Moses's forceful header was then pushed out by De Gea.
Suárez was showing remarkable stamina. Brought down by Jones, Suárez curled the free-kick over the United wall, onto De Gea's bar and over. Rooney responded, striking a good free-kick parried away by Mignolet, who then pushed the Englishman's shot over. As Clattenburg then blew the final whistle, United were in the fourth round (where they host Norwich City), Moyes was smiling and the fans were singing.