Swansea's 5-0 Wembley win over Bradford back in February meant the club picked up the first major honour in their 101-year history.
It also secured passage into the Europa League, giving manager Michael Laudrup an extra competition to consider when it comes to handling his squad this season.
But speaking ahead of Wednesday night's third-round tie at Birmingham, Taylor sees no reason why retaining the trophy should be bottom of their list of priorities.
"We could have said this competition was not important last season, and that staying in the Premier League was the most important thing," said the Wales left-back.
"But games like this build momentum.
"We have to go there, perform as we've been performing. Hopefully Birmingham aren't great on the night and we can get the result."
Taylor missed out on the Wembley action having broke his ankle against Sunderland last September, the injury keeping him out of any Swans action until late April.
The defender then suffered a haematoma in last month's 5-1 victory over Petrolul Ploiesti in Europa League qualifying, and was facing the prospect of another eight weeks out.
However, the 24-year-old has made a quicker-than-expected return and is relieved to be fit to face Blues at St Andrew's in midweek.
"I had a dead leg against Petrolul which would have kept me out for two days and I was going to travel to Tottenham," he said.
"We decided just to leave it for that game and then I'd be back. But on the Sunday, it re-bled and that's just my luck at the moment.
"I could have been out for eight weeks, but luckily it was just over three and it's great to be out here.
"As soon as it happened, I had to go into hospital. It felt like cramp, my leg locked solid and became massive and I was thinking 'what's happened here?'
"It was difficult, I didn't realise how bad it was going to be, but luckily no surgery was required and I hope I have come through it unscathed.
"I was worried, and it wasn't a nice thing to happen because I didn't know what it was.
"I was just unlucky and it was annoying because it was a trauma injury, it's not like a pulled hamstring or something.
"Luckily I've got back quick. Hopefully that is hospitals and I done with for a while."