The 19-year-old kept his nerve to convert the winner from the spot after an epic third-round tie against Queen of the South was decided by a tense shoot-out.
The Jambos - who were drawn against Edinburgh rivals Hibernian in the last eight - now turn their focus back to league business when they host Dundee United on Saturday.
The administration-hit club started the campaign with a 15-point deduction but now sit on minus eight points, with nearest rivals St Mirren having claimed just a single point so far.
Carrick told HeartsTV: "I just want to keep the momentum going and to keep getting goals for the team and hopefully push us up the table more and get some points."
Hearts finally came out on top 4-2 on penalties against Queen of the South, having been pegged back three times over the course of 90 minutes and extra-time.
Carrick eventually netted the winning goal in the shoot-out to ensure Hearts' name would be in the hat for the quarter-final draw.
He said: "It was good to score the goal and it was unbelievable to do it in front of all the fans.
"I was told to go on and try to make something happen.
"To finally put the ball in the back of the net was unbelievable. That's what I was aiming to do."
Meanwhile, Hearts great John Robertson has agreed to remain at the club in a coaching capacity at least until the end of January.
The Jambos' record goalscorer volunteered his services to the financially stricken club in the summer.
He told the Hearts website: "I initially offered my services to help the club through what was a very difficult period.
"I want to do what I can to help Heart of Midlothian Football Club and have agreed to keep coaching here until the end of January.
"I'm really enjoying it and hopefully the players are benefiting from it as well.
"We will hopefully be in a position to further assess things come the end of January."