The Dutch top Group D with 15 points after five matches and host Romania on Tuesday.
In a dull first half, the hosts lacked the necessary urgency to create an opening but a minute after the break Sneijder's replacement, Rafael van der Vaart, netted following good work from Daryl Janmaat.
Janmaat picked up his second assist 19 minutes before time when Robin van Persie scored from close range to double the lead and Ruben Schaken fired home the third after 83 minutes.
"We tried a passing game in the first half although our positional play was not good and our midfielders did not pass accurately enough or did not pass at all," Netherlands coach Louis van Gaal told reporters.
"We set that straight at halftime and then you are lucky as coach when you open the scoring so early in the second half. From that moment on it became easier for us as they could not defend so tightly anymore."
With Sneijder back after missing the last four internationals, Dutch expectations were high but the Galatasaray player never delivered and wasted a chance to open the scoring after 27 minutes when he fired wide from close range.
Minutes later Sneijder limped off with a suspected groin injury but with Van der Vaart on, Netherlands flowed better in attack.
Nevertheless Estonia, who had scored only one goal in their four previous qualifying matches, came close to opening the scoring on the verge of halftime but Konstantin Vassiljev was denied twice by goalkeeper Kenneth Vermeer.
Van der Vaart operated more closely behind central striker Van Persie than Sneijder did and it made a difference for his opening goal.
Van Persie could hardly miss to make it two while Schaken's third goal, a tight angled shot, was the 1,500th for the Dutch team.
Estonia still had chances in the second period with Martin Vunk shooting over 11 minutes before time when he looked well-placed.
Romania snatched a 2-2 draw with Hungary thanks to a last minute Alexandru Chipciu strike behind closed doors at the Ferenc Puskas Stadium.
The result left the neighbouring countries locked on 10 points with Hungary in second place above Romania on goal difference and five points behind leaders the Netherlands.
It was a strange atmosphere on a freezing, windy night after Hungary were ordered by FIFA to play the match in an empty ground after fans chanted anti-Semitic slogans and displayed offensive symbols in a friendly at home to Israel last August.
About 5,000 supporters, monitored by a large police presence, gathered in several popular places in the capital to watch the game on big screens with no incidents reported.
Hungary, who have failed to beat Romania for 32 years, took the lead in the 16th minute when Tamas Hajnal's free kick was headed home by Vilmos Vanczak from seven metres.
Adrian Mutu equalised with a calmly struck penalty in the 68th after Zsolt Korcsmar tripped the former Chelsea striker.
The Hungary defender made amends two minutes later when he was pulled down in the opposing box and Balazs Dzsudzsak converted the spot kick to put his side back in front.
The game seemed destined to end with a home win at last for the Hungarians before substitute striker Chipciu levelled in added time after a scramble in front of goal.
Group leaders Netherlands beat Estonia 3-0 in Amsterdam while fourth-placed Turkey have six points afer winning 2-0 away to bottom side Andorra. The Estonians have three points.
Turkey struck two first-half goals to beat bottom side Andorra 2-0 although an unimpressive performance did little to boost confidence about their prospects of reaching the finals.
The visitors went ahead on the half-hour with a free-kick by Selcuk Inan from the edge of the area that flew into the far corner of Josep Gomes net after Burak Yilmaz was fouled.
Burak Yilmaz doubled Turkey's lead just before the break with a cross-shot from a narrow angle after being put in by Umut Bulut though they failed to capitalise on any more chances.
Turkey, who are struggling in fourth place in the six-team group with six points from five matches, host Hungary on Tuesday having lost the away fixture 3-1 in October.