The score of 3-0 seemed to remain flashing unchanged until the final whistle in the game, but in the three-minute stretch of the added time Russia's Denis Cheryshev upped the score to 4-0 with a swift shot of the ball over the Saudi Arabian goalkeeper.
Cheryshev hammered in a fourth with the outside of his left in stoppage time and Golovin capped a magnificent performance by bending in a last-gasp free-kick.
Amnesty International released a statement after Russian Federation beat Saudi Arabia 5-0 in their opening game condemning Tatchell's arrest as "outrageous".
The World Cup kicks off in Russian Federation on Thursday as years of preparations dogged by diplomatic scandals give way to a month-long feast of action on the field.
"Our goal is to beat Russian Federation". Moody's Investors Service said last month that any boost will be limited and short-lived, well short of the fillip provided by the 2014 Winter Olympic Games in Sochi, which cost a record $50 billion.
That had an effect on Gazinsky's goal, which came after the Saudis struggled to clear a Russian corner, and the header from the tall and stocky Dzyuba.
Cherchesov urged his side to forget the result, thinking ahead to Tuesday against Egypt before a showdown with Group A favourites Uruguay in 10 days time.
Dzagoev pulled up with an injury midway through the first half and was substituted in the 24th minute.
Russia's players were significantly taller and heavier than the Saudis, a help when challenging for the ball and when the penalty area was crowded.
But Russia's win was all but sealed in the 71st minute with Golovin again the creator, the CSKA Moscow midfielder producing a lovely centre from the right that allowed Dzyuba to nod in within two minutes of his introduction.
British LGBT campaigner Peter Tatchell was arrested by Russian Police in Moscow and released on bail on the opening day of the World Cup.
Saudi Arabia had 60 percent of possession and played almost twice as many passes as their hosts, but were still blown away on opening day.
Our Won Jung-hwan is standing by for us near the stadium for the opening match in Moscow.