Shaw returned to the starting line-up after being out for a month with a hamstring injury and slotted in on the left of three centre-backs as Solskjaer deployed a 5-3-2 formation.
Things did not go well at all for Shaw or United as they found themselves 2-0 down in 13 minutes, with the 25-year-old at fault for both goals.
Leipzig netted a third in the second half and late strikes from Bruno Fernandes and Paul Pogba were not enough to save the Red Devils as they were eliminated at the group stage.
Scholes almost felt sorry for Shaw, being thrown into a crucial game after injury and at centre-back when he is more accustomed to playing on the left flank.
‘Luke Shaw, he’s not played football for, what is it, a month? And he’s a lad who needs weeks to get his fitness back,’ Scholes told BT Sport.
The England international was replaced by Brandon Williams on the hour mark, while Solskjaer also had Eric Bailly, Axel Tuanzebe and Timothy Fosu-Mensah as defensive options on the bench.
‘Today we started too passive,’ he said. ‘Not aggressive enough in closing down the crosses or the diagonal balls and, of course, this is something that we prepared for.
‘We knew they were going to put loads of crosses into the box and we just couldn’t deal with them.’
Goals from Angelino, Amadaou Haidara and Justin Kluivert gave Leipzig the victory and send United into the Europa League after finishing third in the group.