Izaguirre started his career at Motagua, where he stayed for seven years. In 2010, Celtic spent £600,000 to bring him to Parkhead. He picked up four major player awards in the 2010–11 season, which was his first season in Scotland.
On 22 July 2008, he had a 17-day trial with English Championship outfit Ipswich Town, but was unable to play in any matches, having not received international clearance from Motagua. He therefore failed to win a contract and the move fell through. In July 2010, it was reported that Toronto FC of Major League Soccer were interested in signing Izaguirre to a designated player contract, after two stand out performances in the CONCACAF Champions League against the Canadian side, but this move did not happen either.
In the 2011 League Cup Final defeat to Rangers, Izaguirre was sent off in extra time for a foul on Vladimír Weiss. On 6 April 2011, he had another man of the match performance against Hibernian in a 3-1 SPL victory, setting up Stokes for the first goal of the game. In the 2010-11 SPL season, Izaguirre achieved a clean sweep of available senior player awards as chosen by players, organisations and the Celtic supporters. Izaguirre won both the SPFA Players' Player of the Year and Scottish Premier League Player of the Year awards for 2010–11. The judging panel for the latter award described Izaguirre as a "class act", while Celtic assistant manager Johan Mjallby commented that Izaguirre could become one of the best players in his position in world football. On 8 May 2011, at Celtic's award ceremony, Izaguirre picked up the Players' Player of the Year Award. Finally, he was voted Scottish Football Writers player of the year, rounding off a highly successful first season in Scottish football for Izaguirre.
Izaguirre started in the Celtic side on the opening day of the 2011–12 season against Hibernian before suffering a broken ankle in the second match of Celtic's season on 7 August against Aberdeen. Izaguirre required surgery on his ankle and was ruled out for between four and six months. On 27 August 2011, he signed a new contract with Celtic, due to run until 2015. Izaguirre returned to training, after recovering from his ankle injury, in December 2011. He made his long-awaited return as a substitute on 2 January 2012 in Celtic's 3-0 win over Dunfermline Athletic at East End Park.
Izaguirre is a left back who plays like an old fashioned wing back. He is most well known for his overlapping runs and ability to fashion attacking moves with other players down the left hand side of the pitch and support forward play. He possesses great pace, is a skillful dribbler and a good crosser of the ball. He is also very composed on the ball when his side are defending and rarely gives away possession. Izaguirre is also a good reader of the game as despite spending a lot of his time attacking, his pace means that he is not caught badly out of position and his attacking play has rarely exposed his current side Celtic defensively. Although he is rather short and slight of build, he is not easily pushed around by opposition players though his height does make him a vulnerable target in the air.