The star forward duo have now both scored in each of Barca's past five matches and Neymar gladly accepted a pass from his partner in crime to net an early opener on Tuesday.
Messi doubled the advantage in the 24th minute - his ninth goal of the season - and the prospect of Luis Enrique's team bouncing back from their surprise 3-2 loss at Paris Saint-Germain last time out in the Champions League was rarely in doubt until substitute Anwar El Ghazi struck in the 88th minute to set up a tense finale.
Ajax pressed for an equaliser, allowing Neymar's replacement Sandro to seal the points on the break.
Barcelona can now look ahead to Saturday's eagerly anticipated La Liga showdown with Real Madrid - where Luis Enrique will finally have the enviable option of combining Luis Suarez with Neymar and Messi in a formidable attacking trio.
Ajax - now without a win in nine away matches in Europe's premier club competition - handed a first Champions League start to left winger Ricardo Kishna, while Barcelona again selected Marc-Andre ter Stegen for European action despite the goalkeeper's shaky showing in Paris.
Messi started the night three goals shy of Raul's all-time record of 71 Champions League goals, but the mercurial Argentinian played the role of provider in the seventh minute.
Niki Zimling struck an aimless pass into midfield and Andres Iniesta needed little invitation to thread a trademark throughball to Messi, whose shot squeezed home despite goalkeeper Jasper Cillessen’s best efforts.
Iniesta sauntered through the Ajax defence six minutes before half-time, only for Cillessen to deny him, and Ajax almost had a lifeline before the break when Kolbeinn Sigthorsson's shot spun wide having taken a deflection off Gerard Pique.
Barcelona resumed on the front foot and Messi prodded the ball into the net from an offside position when Cillessen parried Pedro's well-stuck 49th-minute drive, before the Barca talisman weaved through several challenges to shoot into the side netting.
Ricardo van Rhijn then sent a low effort narrowly wide from Lasse Schone's pass and Ajax were offered another moment of encouragement in the 63rd minute when Pique nervously sliced a left-wing cross from Arkadiusz Milik onto the roof of his own net.
Either side of that scare, Neymar and Messi departed to standing ovations but Barcelona appeared to shift focus to their weekend engagement as they closed the match in disjointed fashion.
El Ghazi bundled the ball home from close range and then Jordi Alba was forced into a desperate challenge to stop the teenage winger from levelling matters, before Sandro scored his vital third goal of the season in stoppage time.
