The Blades take on Nottingham Forest in the first leg of their semi-final at Bramall Lane on Saturday and Heckingbottom said his side will never be forgotten by their fans if they can secure promotion.
Heckingbottom said: “Successful teams, you remember them; big moments, you remember them; and the players who supply those big moments, the fans remember them.
“The players do themselves, it’s the highlight of their careers. We told the players, the chances are you’ll have a picture of a big moment, a promotion, up on your wall.”
The Blades, bidding to bounce straight back to the Premier League following relegation last season, sat in 16th place when Heckingbottom replaced Slavisa Jokanovic in November.
They secured a fifth-placed finish behind Forest on the final day with a thumping 4-0 win against Fulham at Bramall Lane and the winner of their two-legged tie will play either Huddersfield or Luton at Wembley.
Heckingbottom said: “The prize is there. The semi-final after 46 rounds and that’s what it is. Everything is magnified, the feeling, the build-up, the intensity.
“That’s what the play-offs provide. They magnify everything. There’s a bigger audience on you because everything else is finished.
“These moments can be extra special, but likewise, if they don’t go your way, they hurt.
“We’re down to four. One of these team is going to be in the Premier League. We know two who are already there, now one from four.”
The Blades announced key defender Chris Basham has followed fans’ favourite Billy Sharp in committing himself to the club by signing a new two-year deal on Thursday.
Sharp, the Championship’s all-time leading goalscorer, signed his new contract last month, but Heckingbottom said he could not confirm new deals for several other players.
The former Leeds boss said he was still waiting to learn what his playing budget will be as he looks to build on his successful first season in charge later this summer.
Heckingbottom added: “I’m pretty clear on it. I’m not fighting for money, I’m just fighting for the ability to make the best decisions with it.
“We are the ones on the ground and know the dynamic of the squad. You always want to improve the squad and also improve the processes of how things are done and that’s a big part of moving forward.”
Heckingbottom has no new injuries following last week’s win against Fulham, with Morgan Gibbs-White expected to start after he was withdrawn late on in that match.
Sharp is hoping to feature at some stage in the play-offs as he works his way back from a long-term calf injury, while George Baldock is in contention having returned to the bench last week following a hamstring strain.