Wembley, which was already scheduled to host the semi-finals and final, will now replace the stadium and stage three group games and one last 16 match.
UEFA made a decision to reallocate the three group-stage games and one round-of-16 game originally given to the Belgian capital because of delays in the project to build a new 60,000-capacity Eurostadium on the site of Heysel.
The decision means that Wembley will stage seven games in total - the final, both semi-finals, a last-16 match and three group games - resulting in the biggest festival of global football in the country since Euro 96.
A draw was also held to decide which cities would host games from the various groups.
"Missing out on Euro 2020 does not necessarily mean a death blow for our stadium plans", he said.
"Uefa ranked the Champions League final as one of their best events and praised the way in which the Cardiff 2017 local organising committee delivered the showpiece".
"We discussed with Brussels for quite a long time and they were not able to provide us with all the documentation".
While the full schedule is still to be confirmed by UEFA, the Stadio Olimpico in Rome will stage the opening game.
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"We received a letter yesterday saying that there will be a final decision on January 31".
"Experts from (our) administration felt it was a high risk for UEFA to wait".
The Football Association of Wales (FAW) said they were "extremely disappointed" that their bid to host four matches at Cardiff's Principality Stadium had been unsuccessful.
Should the Republic of Ireland qualify for Euro 2020, they will play one of their group games in Bilbao.
The FAW statement read: "The concept of taking UEFA EURO 2020 to 13 different countries was devised to allow smaller countries, like Wales, to have a unique opportunity of being involved in staging a major tournament".
The quarter-finals and three group matches will be held in Munich (Germany), Baku (Azerbaijan), Rome (Italy) and St Petersburg (Russia).
England's national stadium has been paired with Glasgow to host the six matches in Group D.