The Manchester United forward overtook the ex-Arsenal legend in the all-time goalscoring list for Oranje and is now behind only Patrick Kluivert
The visitors have never scored against the Dutch, but belied their minnow status with a committed first-half display, with the game goalless at the break.
But Van Persie dragged his team to Brazil 2014 with two goals in four second-half minutes that, coupled with Romania's defeat at home to Turkey, sealed their spot at the top of Group D.
Louis van Gaal made just two changes to the side that drew 2-2 with Estonia on Friday as Ron Vlaar and Ruben Schaken replaced Bruno Martins and Arjen Robben.
Wesley Sneijder looked to catch home goalkeeper Ferran Pol out of position with a fourth-minute free-kick, but his effort from 30 yards just bobbled wide.
The hosts acquitted themselves well, however, with Marc Pujol curling a free-kick wide of Michel Vorm's goal and then floating another dangerous set-piece into the penalty area, which was flicked just wide by Andorra's record goalscorer Aldefons Lima, who missed out on his eighth international strike.
Netherlands dominated possession, but they could not carve open a stubborn Andorra defence with regularity.
Sneijder was heavily involved, drilling a low effort straight at Pol before heading Daryl Janmaat's cross just wide.
Ivan Lorenzo troubled Vorm just after the half hour with a low drive that the Swansea City man spilled, but recovered to grab with Pujol set to pounce.
Soon after, Van Persie flashed a free-kick wide of goal and the Dutch continued to be frustrated as the sides went in at the break level.
With strong words from Van Gaal no doubt ringing in their ears, the Dutch came out after the break with a renewed purpose and they finally took the lead five minutes into the half.
Kevin Strootman drifted onto the right flank before cutting the ball to Van Persie on the edge of the box and the Manchester United man curled a sumptuous effort beyond Pol.
Netherlands then doubled their advantage just four minutes later, again through Van Persie, who pounced after Janmaat's cross had been poorly parried into the six-yard box by the home keeper.
The goal was his 38th for the Netherlands and takes him above Dennis Bergkamp on his nation's all-time top goalscorers list in second place, with only Patrick Kluivert now above him on 40.
Van Persie's tail was up and he went in search of the match ball, but sliced a right-footed effort wide after 67 minutes before heading Schaken's centre wide shortly after.
However, he was unable to add to his tally and Oranje cruised to a victory that now allows them to make plans for Brazil.