Breel Embolo put Switzerland in front after 49 minutes, his close-range effort full reward for their superiority.
But Moore claimed his sixth international goal 16 minutes from time – and Wales were relieved when Mario Gavranovic’s late effort was disallowed for offside following a VAR check.
A point was more than Wales deserved for large parts of a contest in which they were often second best and key men Gareth Bale and Aaron Ramsey were kept quiet.
But Moore’s header will give Wales – semi-finalists at the last European Championship in France five years ago – belief that they can progress from a competitive group.
The result will place greater emphasis on Wales’ next game against Turkey, who lost their opening Group A fixture 3-0 to Italy, in Baku on Wednesday.
Bale, Wales’ record goalscorer, had not scored for 11 games going into the match – a run stretching back to October 2019.
But he had netted in his two previous appearances against Switzerland, with Ramsey also on target when the two sides last met in a 2-0 Wales victory in 2011.