Ruud van Nistelrooy, David Beckham and Paul Scholes all struck inside 21 minutes to overturn the Hungarians' first-leg lead with something to spare.
Then, after former Charlton keeper Sasa Ilic had been sent off, van Nistelrooy struck home a penalty before Ole Gunnar Solskjaer wrapped up the win.
It didn't come without worry though, as Scholes was stretchered off just after the break with a nasty-looking ankle injury.
United suffered a problem inside the first 20 seconds when van Nistelrooy jumped to contest a long ball from Veron and landed awkwardly.
The Dutchman gingerly regained his feet, only to be clattered fairly by Otto Vincze.
Ilic's poor clearance kick gave Beckham the chance to launch an attack, although the former Charlton man redeemed himself with a flying save from Scholes's volley.
It counted for nothing though, as the offside flag had already been raised.
Boss Alex Ferguson had challenged his team to match the tempo they showed on Friday and they certainly started quickly.
Their opponents were playing some neat stuff too, but, within six minutes, their lead had gone.
Phil Neville blamed for the Hungarians' winner in Budapest fed a ball forward, which Scholes touched through to van Nistelrooy.
The Dutchman sped clear of a static ZTE defence and slid the ball under Ilic to level the aggregate score.
Beckham then fired over before Scholes failed to hit the target with an acrobatic volley after reaching van Nistelrooy's cross at the near post.
Tamas Szamosi let fly with an ambitious attempt at the other end as the visitors tried to relieve the siege on their own goal.
However, the respite was brief and when United were awarded a free-kick five yards outside the Hungarian box, Beckham strode forward and stroked the ball past Ilic to put his team ahead in the tie.
ZTE were struggling badly, Ryan Giggs floating another free-kick across the area and just out of the home attack's range.
Ilic saved brilliantly with his feet from Giggs after Scholes's curling pass across the area had picked out the Welsh winger.
United wiped away the worry of conceding an away goal after 21 minutes when Scholes miss-hit a first-time shot from Beckham's cross.
The ball stuck under the England man's feet, and with the ZTE defence committed, he placed it neatly past Ilic for number three.
Ilic had little chance with any of the goals, but produced a great stop to deny Giggs after Igor Budisa's poor clearance had fallen straight into the Welshman's path.
Beckham showed his eagerness for the battle, chasing back to deny Krisztian Kenesi after Neville had advanced into a forward position.
United had eased off the gas, but Giggs still almost latched on to a lofted Veron through-ball.
Beckham switched to the left, with Scholes moving to the right and Giggs striding forward down the middle, although the next goalmouth action was at the other end, Roy Carroll watching carefully before gathering Kenesi's free-kick after it had sped through a host of bodies.
Van Nistelrooy took possession in a dangerous position on the left-hand side of the area but failed to pick out a team-mate with his cross. ZTE were enjoying their most productive spell, Vincze firing over from long range.
Having already been denied twice, Giggs opted for a square ball when van Nistelrooy sent him clear in the area. The angle may have been tight, but the Welshman should have shot rather than squander possession altogether, playing the pass into an area no home player had found. Rio Ferdinand was taken off to rapturous applause midway through the second period as O'Shea was given another chance to impress.
Van Nistelrooy fired over from a good position after shaking off his marker, then Veron's speculative 40-yard effort fizzed past Ilic but wide of the post.
Ilic raced from his goal the next time the Dutchman sprinted clear, hauling Van Nistelrooy down and leaving the referee with little option other than to award the penalty and send the unfortunate Yugoslav off.
Ilic took the news badly and with reserve keeper Geza Turi replacing Ferenc Babati, it was two minutes before van Nistelrooy smashed the ball home for number four.
The crowd had been urging Diego Forlan to take it, given the Argentinian's failure to find the net in a competitive game for the Old Trafford club.
Forlan thought he had broken his duck with a diving header eight minutes from time, only for the celebrations to be cut short after a linesman ruled the ball had previously gone out of play.
As if to rub salt into the wound, Solskjaer almost immediately netted another, finding the far corner with ease after racing into the area.
Man Utd: Carroll, Phil Neville, Ferdinand, Blanc, Silvestre, Beckham, Keane, Veron, Scholes, Giggs, van Nistelrooy.
Zalaegerszeg: Ilic, Csoka, Urban, Budisa, Szamosi, Babati, Darko Ljubojevic, Molnar, Vincze, Kenesei, Egressy.