The 24-year-old is currently on loan at Upton Park from Liverpool but the beginning of his West Ham career has been blighted by hamstring and knee injuries.
In between setbacks the England striker had netted just once for the Hammers, that goal coming in a 3-1 defeat at Tottenham in November.
Carroll is now fit but he admits it was difficult to settle into Sam Allardyce's side due to his injuries.
"Coming into a new team and getting injured in my first game to put me out for six weeks was frustrating," he told West Ham's official website.
"Then I was back for a couple of games and was out for eight weeks. It's hard when you're injured anyway, but when it's at a new club it's even harder.
"I've kept my head up as much as possible and worked hard with the physios and fitness coaches to get back and I showed on Saturday that I am fit."
Carroll praised the club's medical team for getting him back to fitness in time to make a match-winning start against Swansea as West Ham claimed only their second Barclays Premier League win in 10 games.
He said: "I've been keeping fit in the gym and all the backroom staff have been helping me. The fitness coaches and physios have been on me, working me hard.
"When I couldn't run I was in the gym and the hard work paid off with 90 minutes on Saturday.
"I didn't find out until the day of the game that I was going to start and obviously I was over the moon with the decision. I just had to work hard, run around and keep the ball as much as I could.
"It was a tough game and I'm just happy that we got the three points."
Carroll's 77th-minute header was the first time he had scored at Upton Park and he revealed he was delighted to be off the mark in front of the Hammers' supporters.
"It's a great feeling to score anywhere, but when you've got the fans behind you like this, singing your name after you've scored, it's even better," he said.
"Coming off the pitch, man of the match and with three points, it's a great feeling. The fans are great, they got behind us and they were like the 12th man, helping us along.
"Even when Swansea put the pressure on late in the game, they gave us confidence with the singing and cheering us on. It's great."
West Ham now sit 11th in the league table and Carroll's winner took them nine points clear of the relegation zone after a slow start to 2013 saw them drifting towards the bottom three.