Solskjaer's side slumped to an embarrassing 2-0 defeat at lowly Watford last weekend to raise fresh questions about the United manager's ability to inspire his players.
When Matty Longstaff put Newcastle ahead in the first half, Solskjaer's critics were sharpening their knives again.
But Martial rode to Solskjaer's rescue with the equaliser and Greenwood's thunderbolt put United ahead before Marcus Rashford rounded off a first half blitz.
Martial netted again after the interval as United ended a two-match winless run in the Premier League and moved into seventh place, just four points behind fourth placed Chelsea in the race to qualify for the Champions League.