Praise be to Allah.
Yes, this method is sufficient. The scholars are unanimously agreed that what is obligatory in wudoo’ is to wash each part once, although washing each part two or three times is Sunnah. That is because of the report narrated by al-Bukhaari (157) from Ibn ‘Abbaas (may Allaah be pleased with him) who said: The Prophet (peace and blessings of Allaah be upon him) did wudoo’ washing each part once.
Al-Nawawi (may Allaah have mercy on him) said: The Muslims are unanimously agreed that what is obligatory is to wash each part once, and that washing each part three times is Sunnah. There are saheeh ahaadeeth which mention washing each part once, and each part three times, and washing some parts three times, some two times and some once. The scholars said: the difference indicates that all of these are permissible, and that washing each part three times is most perfect and washing each part once is sufficient. This is how the variations in the ahaadeeth are to be interpreted. End quote.
Al-Shawkaani (may Allaah have mercy on him) said: The saheeh ahaadeeth speak of washing each part once, twice or three times, or washing some parts three times and some twice. The differences indicate that all of these are permissible, and that washing three times is most perfect and washing once is sufficient. End quote from Nayl al-Awtaar, 1/188.
But we should not persist in washing each part only once and not do that which is more perfect, because by doing so we are depriving ourselves of more reward.
And Allaah knows best.