United lost the shoot-out 11-10 after David De Gea missed the decisive kick following a 1-1 draw in Gdansk.
Rashford has been carrying injuries to his shoulder and left foot for much of the season but the 23-year-old said many of his teammates were also playing through the pain.
"I could walk you into the dressing room, I could show you five, six, seven players and me as well who had injuries here, from September, the beginning of the season," Rashford said.
"We all stayed together as a unit and we fought to be successful for the club.
"This year it didn't happen, but next year, we have to go away now and clear our heads and when we come back we start fresh. When Ole [Gunnar Solskjaer] came in, there was a process and the players believe in this process and this isn't the end of the process."
United end the season having finished second in the Premier League, reached the final of the Europa League and the semifinals of the Carabao Cup. However, Rashford was quick to dismiss the suggestion that the campaign has been a successful one.
"I don't want to hear: 'Oh, they were so close' because it doesn't mean anything. There's one winner and there's one loser and today we lost and we have to find out why and make sure that next time we don't lose.
"In the league we finished second, second doesn't count for anything. It counts for nothing. Manchester City won the league, and we finished second but it means nothing. Villarreal win the Europa League and we finish second in it. For us it's nothing.
"We come next season with more desire, more hunger and we have to do our best on the pitch. I promise we are close, but close is not good enough, we have to be there."
Rashford said he been subjected to "at least 70 racial slurs" on social media following Wednesday's defeat by Villarreal in the Europa League final.