United, who beat Perth Glory 2-0 at the weekend, almost made the breakthrough in the second minute when a fine run and pass from Victor Lindelof freed Rashford inside the Leeds box, but the England international saw his low strike hit the post.
Paul Pogba - now likely to remain at Old Trafford - also played a huge role in the breakthrough, though, with the midfielder's clever pass freeing Wan-Bissaka, who was a constant threat down the right in the first half of action.
Pogba sent a free kick just wide of Kiko Casilla's post in the 14th minute, before the French midfielder forced the Leeds goalkeeper into a smart save moments later as chances continued to arrive.
Bielsa's side were showing signs of improvement as the first half developed, though, and on-loan Manchester City winger Jack Harrison sent one into the side-netting in the 22nd minute following a smart move.
Pogba again missed the goal with a strike two minutes later as the France international continued to impress in a midfield two alongside the energetic Scott McTominay.
Chances were arriving at both ends, though, and Romero had to keep out attempts from Hernandez and Patrick Bamford in quick succession as Leeds - without Premier League football since 2004 - continued to show signs of improvement.
United doubled their lead in the 27th minute, however, when a quick counter-attack ended with McTominay feeding Rashford, who produced a wonderful piece of skill before expertly picking out the bottom corner of the net.
There were more opportunities late in the first period, with Hernandez again coming close for Leeds before Romero had to keep out a header from Kemar Roofe at the far post. Rashford also forced Casilla into another smart save in the 39th minute during an exciting end to an entertaining half of football.
The final chance of the first 45 minutes came for James after Rashford had found his teammate inside the Leeds box, but the summer arrival from Swansea City saw his low effort clip the outside of the Leeds post.
As expected, Solskjaer sent an entirely new XI onto the field for the second period, with Martial at the tip of the attack and the likes of Tahith Chong and Angel Gomes also given opportunities to impress.
The first chance of the second half actually fell to Leeds in the shape of Harrison, but the attacker could only poke his effort wide of Joel Pereira's goal from close range.
United scored their third in the 51st minute, however, when Jones climbed highest inside the penalty area to head a brilliant Andreas Pereira corner into the back of the net.
Casilla had actually made a brilliant save to prevent an Andreas Pereira effort finding the back of the net moments before, but the former Real Madrid goalkeeper could do nothing from the resulting corner.
Harrison scooped another effort over the United crossbar in the 59th minute, before Gomes missed the post down the other end during an open period of the match.
The Red Devils scored their fourth from the penalty spot in the 69th minute, with Martial stepping forward to convert after a wonderful piece of skill from Chong had led to Liam Cooper chopping the youngster down.
Jesse Lingard sent a low effort just wide of the Leeds post late on as United chased a fifth, but the team in red ultimately declared at four following an impressive 90 minutes of football.
Solskjaer's team will now travel to Singapore to face Inter Milan on Saturday, while Leeds, who have only brought a 16-man squad Down Under, will play their second and final match in Australia against Western Sydney Wanderers on the same day.
MANCHESTER UNITED FIRST-HALF XI: Romero; Wan-Bissaka, Lindelof, Bailly, Rojo; McTominay, Pogba; Greenwood, Mata, James; Rashford
MANCHESTER UNITED SECOND-HALF XI: J Pereira; Dalot, Smalling, Jones, Young; Matic (Garner 78'), Pereira; Chong, Gomes, Lingard; Martial
LEEDS UNITED: Casilla; Dallas (Hosannah 66'), Cooper, Berardi, Douglas; Phillips, Forshaw, Harrison; Hernandez (Stevens 77'), Bamford (Bogusz 63'), Roofe