Dirk Kuyt's late winner secured the Reds' progression after Park Ji-sung had cancelled out Daniel Agger's opener.
"[We didn't deserve to win] on possession, and probably not on general play," Gerrard told ITV.
"You've got to give Manchester United credit, they controlled large periods of the game today. But the most important thing was the result, and we got their in the end.
"We had a really tough match in the week so you've got to give our team a lot of credit for the effort and commitment we put in today."
Gerrard also hailed the performance of striker Andy Carroll, who, having been on the receiving end of plenty of criticism since moving to Anfield, supplied the crucial flick-on in the build-up to Kuyt's winning goal.
"He's a good player and he's very important to Liverpool," he said. "If it wasn't for Andy today, maybe we wouldn't be in the draw tomorrow.
"We played to Andy Carroll's strengths. He can cause any team a lot of problems. Dirk [Kuyt] got the winner today, and Daniel Agger got the first, but Andy Carroll made a massive contribution."
The 31-year-old also praised the role of the Liverpool fans in driving the team to victory following their tiring League Cup semi-final second leg against Manchester City on Wednesday evening.
"I thought the Liverpool fans were superb today, like they were in midweek," Gerrard said.
"They never let us down. It was a big game, and they played a part today, they pushed us through. I think they realised that we were very tired from midweek, but they kept pushing us."
Substitute Kuyt added that the result, as well as Liverpool's progression to the League Cup final, showed they could compete with the country's top sides.
"It was a great header from Andy, and I just ran [onto it]," he said. "The only thing to do was just to score.
"We proved that we can compete with the best. Maybe they were the better team, and they were desperate to win, but at the end of the day it's about scoring goals. Whoever scores the most goals wins the game, and it was up to us."
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