Gareth Southgate's side were in danger of a damaging defeat in their bid to qualify for next year's World Cup when Slovakia snatched a shock early lead at Wembley on Monday.
It was a mistake from Rashford, giving the ball away and then failing to track back, that paved the way for Stanislav Lobotka to score.
But the 19-year-old, making his first England start at Wembley, showed maturity beyond his tender years and refused to be cowed by his blunder.
Rashford wasn't finished there and he capped his man of the match display with a superb long-range strike in the second half to put England on the brink of qualifying for the finals in Russia.
Dier was impressed by Rashford's performance, but the Tottenham midfielder believes it is far too soon to crown his team-mate as England's new leading man.
"He's got fantastic potential like a lot of young English players," Dier said.
"But I'm going to be the last person to big them up to let someone else knock them down.
"I'm sure he will keep his head down, keep working hard and hopefully he can fulfil that potential."
Although Rashford is a novice on the international stage, he already looks comfortable in the harsh spotlight trained relentlessly on the England team.
So, after he made that painful early error, Dier insisted he wasn't worried about Rashford losing confidence.
"I don't think you need to say anything. Marcus is smart enough," he said.
"He is a great lad, very level-headed. It was a great show of character from him, to come back after that and get an assist and a goal."
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Rashford's decisive interventions took the attention away from England's first-half travails, a spluttering spell that had many fearing Southgate's men are still light years from winning a major trophy.
Under the management of Roy Hodgson, England's last tournament ended in humiliation at the hands of minnows Iceland at Euro 2016.
But Dier is adamant his team's comeback against Slovakia showed they are mentally stronger now.
"It's not the same stage but it does show a progression. We went behind but the most important thing for us was not to make any more rash decisions and concede again," he said.
"The important thing is to stay calm and not hit the panic button. We did that and at half-time we were really confident we would get the win.
"In every game, no matter what, we need to have that attitude.
"It's easier said than done and the bigger the stage the harder it is to do it.
"This was a stepping stone in the right direction."
Dier was also encouraged that England's training ground preparation paid off as they effectively smothered Slovakia to close out the win.
"It wasn't particularly pretty at times. We sat off in our own half and they didn't create anything from that.
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