The Red Devils struggled to open up an ultra-defensive Astana for long periods of the contest, but 17-year-old Greenwood came up with a brilliant finish in the 73rd minute to decide the European clash.
Diogo Dalot, Phil Jones, Marcos Rojo and Fred were also handed rare starts for the Red Devils as Solskjaer shuffled his pack at Old Trafford ahead of Sunday's trip to West Ham United in the Premier League.
United came close to making the breakthrough in the third minute when Fred struck one from distance, but the midfielder's powerful attempt clipped the crossbar before dropping behind for a goal kick.
Rashford had a golden chance to make the breakthrough in the 13th minute when the ball fell kindly for the England international following a corner, but Eric was on hand to make a brilliant stop.
Greenwood was next to come close in the 16th minute as the 17-year-old just missed the far post with a powerful strike following a smart flick from Rashford, who was perhaps a surprise starter in the final third when considering his volume of football in the early stages of the season.
Rashford had another opportunity in the 20th minute when he met a clever pass from Rojo, but once again the away side's 37-year-old goalkeeper was on hand to make the save.
Solskjaer's team continued to dominate the ball as the half progressed, but the opportunities did dry up in the latter stages of the first period with Astana happy enough to defend their own penalty box.
The visitors had a half-chance of their own in the 37th minute when a breakaway led to Roman Murtazayev having a clear strike on goal, but the 26-year-old's weak attempt was easily saved by Sergio Romero, who was also enjoying his first minutes of the 2019-20 campaign.
Rashford had the final attempt of the half as he looked to pick out the back of the net with a clever finish, but the attacker's effort was saved by Eric, and the two teams headed down the tunnel at 0-0.
United had another huge chance to take the lead early in the second period when Rojo flashed a brilliant cross towards the far post, but neither Greenwood nor Chong could make contact as Astana, who brought on two substitutes at the break, survived a very dangerous moment.
The Red Devils continued to force the issue as the half developed, and the corner count was rising as Astana were forced to deal with wave after wave of attacks from the English giants.
Rashford sent a dipping free kick just over the crossbar in the 55th minute, while Rojo's driving runs forward and dangerous deliveries were becoming a key feature of the home side's work.
Fred was next to come close for the hosts with the Brazilian seeing his curling free kick kept out by Eric as the score stayed level heading into the final 30 minutes of action at Old Trafford.
The pattern of the match did not change as the hosts continued to dominate the possession, but Astana were defending the width of their own box in effective fashion.
United finally made the breakthrough in the 73rd minute, however, when Greenwood worked some space inside the Astana box before picking out the back of the net with a clever finish through the goalkeeper's legs for his first senior goal.
Eric made a wonderful save to keep out a powerful Rojo drive as the home side looked for a second, with Lingard also rattling the post from distance before Dalot missed from close range.
The 2016-17 Europa League winners did not find a second on home soil, but it mattered little in terms of the final result with Greenwood's goal enough to secure all three points.
Next up for United is a trip to West Ham in the Premier League on Sunday afternoon, while Astana will continue their league campaign away to Zhetysu on the same day.
MANCHESTER UNITED (4-2-3-1): Romero; Dalot, Tuanzebe, Jones, Rojo (Young 79'); Matic, Fred; Chong (Mata 68'), Gomes (Lingard 68'), Greenwood; Rashford
ASTANA (4-4-2): Eric; Rukavina, Tomasevic, Postnikov, Shomko; Tomasov, Maevski, Simunovic (Logvinenko 46'), Sigurjonsson; Rotariu (Mubele 82'), Murtazayev (Janga 46')body check tags ::