Both former United players were going for the job, with Carrick being chosen on Tuesday.
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BBC Sport says Solskjaer, who had three years in charge of United between 2018 and 2021, holds no grudge and is backing Carrick to be a success in the role.
Solskjaer views United's improvement as 'the most important thing' and feels Carrick will do an 'excellent job' at the helm.
Carrick steps in after the dismissal of Ruben Amorim last Monday following 14 months in charge. The former Middlesbrough manager's first challenge is this weekend against Manchester City in the local derby.