The Hungarian stopper was part of the successful reserve team of 2007 and signed in August 2008 after completing a one year loan at Liverpool. Gulasci impressed in Liverpool's reserves and signed a new four-year contract in 2009. In January 2009 he was loaned to Hereford United and made eighteen Football League appearances. In April 2010 the young goalkeeper was again loaned out, this time to Tranmere Rovers, initially on an emergency seven-day deal, and made his debut against Exeter City the following day. The loan was then extended, allowing Gulasci to play in another four League One matches by the end of the season. He again moved to Tranmere Rovers on 17 September 2010. An 'emergency' loan-deal was ratified by the Football League because Tranmere's two experienced goalkeepers were both out of action because of injury. Rovers were happy to have him back at Prenton Park because he had made a big contribution to them retaining their League One status in his previous loan spell there. After returning to Anfield, Gulasci became the clear choice as back-up to Pepe Reina but the Spaniard's excellent form and fitness meant that Gulacsi did not see a minute of first-team action despite being named in the matchday squad twenty-three times by Kenny Dalglish following four earlier appearances on the bench under Roy Hodgson.

Four days before Liverpool were due to visit Hull City for a pre-season friendly in July 2011, Gulacsi joined the Humberside club on loan for the whole of the 2011/12 season. At the same time as announcing this loan move Liverpool confirmed that Gulasci had signed an extension to his contract. In the wake of Liverpool's goalkeeping problems as both Reina and Doni were both suspended, was recalled from Hull ahead of the club's FA Cup semi-final date with Everton at Wembley in April 2012 where he was a reserve to Brad Jones. Gulacsi was named as Liverpool's substitute goalkeeper eighteen times in 2012/13, mostly for Reina but also for Jones when Reina was unavailable. Without ever making a competitive appearance for the Reds the goalkeeper signed a four-year deal with Red Bull Salzburg at the end of the first week of June 2013.

After two years in Austria where he played 65 times he was transferred to German second tier club Rasen Ball Sport Leipzig, who are coincidentally sponsored by Red Bull, but cannot be called that due to German football naming regulations. He began as the second choice keeper, even spending a short time on loan with the reserves in the fourth tier. However in the second half of the season he was given an opportunity due to injury and played fourteen times as they were promoted to the Bundesliga. He was included in Hungary's European Championship squad.

Despite the move up in standard, Gulacsi has now secured his status as RB Leipzig's first choice keeper, appearing in 33 of the 34 Bundesliga games in both the 2016/17 and 2017/18 seasons, the latter of which saw them compete in the Champions League. In 2018/19 he was first choice Bundesliga keeper but rotated for Europa League ties. The following season he was first choice in both the Bundesliga and Champions League, where they reached the semi finals. In 2020/21 Gulacsi played in both legs of RB Leipzig's Champions League round of 16 tie with Liverpool. The games were played behind closed doors in Budapest due to Covid 19 restrictions that made travel between England and Germany impractical. The following season he captained RB Leipzig to their first trophy, the DFP Pokal but missed much of 2022/23 due to a cruciate ligament injury. Gulacsi had to bide his time in 2023/24 when he was not fit in time for the start of the season then had to settle for a place on the bench. He regained his position as number one keeper in February and played in all of Hungary's three games during Euro 2024.

Gulacsi began the 2024/25 season as RB Leipzig’s first-choice goalkeeper. At 35, he played a central role in the Bundesliga campaign, starting every league match and providing stability at the back. His leadership was a key factor in organizing the defence, while his shot-stopping and decision-making helped Leipzig maintain a strong defensive record. Leipzig finished 7th, missing out on European competition for the first time in seven years. By May 2025, Gulacsi decided to retire from international football, concluding over a decade with the national team during which he earned 58 caps, and represented his country at three European Championships.