Financial terms of the deal were not disclosed but British media reported the new contract could see Rashford earn around 250,000 pounds ($316,550) a week as manager Ole Gunnar Solskjaer plans to build his attack around the academy graduate.
"There's nothing I want more than to see the club winning the Premier League again, winning the Champions League again," Rashford told The Times.
"Everybody here deserves much, much more than what we're giving them right now, not only the fans, the club itself. They don't deserve to be where they are now.
"As players, we're the ones who have the opportunity to try and forget about the past five, six years and put the club back on track."
Rashford said he is constantly trying to develop his game and is taking tips from Solskjaer, who was a striker himself.
"That's what drives me every day. If I didn't believe it, I wouldn't be waking up every morning and giving 100% to try and put the club back where they belong," he said.
"The players have faith in the manager now. It can take two or three seasons but I feel we're ready to take the first step."
England international Rashford was Solskjaer's preferred centre forward ahead of Romelu Lukaku last season as he finished with 13 goals in 47 appearances across all competitions.
"It hurts every day to see where we are now, (compared) to where we've been in the past. It makes me more determined to put the club back where they belong," Rashford said.
United begin their new league campaign against Chelsea at Old Trafford on Aug. 11.
(Reporting by Arjun Panchadar in Bengaluru; editing by Sudipto Ganguly)
