The abuse of Rashford came after United midfielder Paul Pogba was also subjected to racist taunts on social media following his failure to score a penalty in Monday's 1-1 draw at Wolves.
Pogba had taken penalties duties from Rashford, who resumed the job against Palace, only to miss from the spot for the first time in his career.
"My response is the same as before the weekend," said Solskajer, when informed of what had happened to Rashford.
"I am lost for words if it keeps going. We keep having all these campaigns, saying no to racism, but these people keep hiding behind fake identities. It is crazy that we keep talking about this in 2019."
United fans will back Pogba after racist abuse says Solskjaer
The racist incidents provided a nasty conclusion to a bad afternoon for United as they lost their unbeaten start to the season thanks to a stoppage-time winner from Patrick van Aanholt, which delivered a first league success over United for 28 years for the Eagles.
For Solskjaer, there was only frustration as United now have just one win from their first three games, a record that falls short of the Norwegian's expectations.
"We have to learn and we have to learn quickly. The last two games we should have taken six points from them if you had been watching as a neutral," he said.
"Of course, we are two penalty misses away from sitting here smiling.
"After half an hour of no danger, one long kick and we didn't narrow down. Then for the last goal, after having all that pressure in the second half and David (de Gea) had not touched the ball, they scored again."
After United had dominated the early stages, Palace scored with their first attack.
Jeffrey Schlupp beat Victor Lindelof to a long clearance from goalkeeper Vicente Guaita, leaving Jordan Ayew to race through and slot it past De Gea to equal his tally of Premier League goals from last season.
Twitter agree to meet with Manchester United over Pogba abuse
United looked to have run out of time until young Welsh winger Daniel James, who had earlier been booked for diving, produced an 89th minute equaliser when he produced a lovely curling finish into the top corner after some sharp interplay had finally prised an opening in the Palace rearguard.
Yet with the home crowd urging their team on to grab a winner during five minutes of stoppage-time, they were left stunned as Palace delivered the killer blow.
Pogba was guilty of losing possession and the ball was worked towards van Aanholt who had charged forward from left-back and although his shot was well struck, it squirmed under De Gea's body into the net.
"I shall certainly remind myself about this one for a few days, until I start thinking about Colchester in the League Cup and then Aston Villa," Palace boss Roy Hodgson said.
There was also VAR frustration for Solskjaer as he believed his team were denied two further penalties, while Palace defender Gary Cahill was only given a yellow card when he brought down Anthony Martial when he appeared to be the last man just outside the area.
"We know these five days have been disappointing, one point instead of six, so we need to stick together and prepare for Southampton next Saturday," he said.
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