The veteran was proud to see Louis van Gaal's fight back against West Ham to stay in the race for FA Cup glory but warned his team-mates they still need to improve
Dimitri Payet gifted West Ham the lead with a well-taken free-kick, but Anthony Martial eventually levelled the scoring as it ended 1-1 at Old Trafford.
A replay at Upton Park will now decide who will progress to the semi-finals, and Carrick is delighted United are still in the race to win a trophy.
"We were well aware of the importance of this match going into the game and wanted to go through in this competition. We haven't quite done it yet but we live to fight another day," Carrick told BT Sport.
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"We'll take the draw in the end, but we were pressing for a winner and definitely wanted to win the game. I always felt we would get back in it but we just couldn't get a winner. Of course there's relief, it's a big opportunity.
"It's the FA Cup and we all want to go far in this competition. To go 1-0 down in the second half and then be pushing for a winner is positive.
"It was a great free-kick [from Payet] but I thought we showed character to get back into the game and push for a winner. We were building pressure and probably had more opportunities.
"The final pass didn’t quite happen for us and we didn’t work the goalkeeper enough.
"It’s the FA Cup, it was end-to-end at times and there was a lot happening in the box but I thought it was a good cup tie.
"With penalty decisions, sometimes you get them and sometimes you don't, so one goal each way is probably fair I think."