Luis Enrique's side took advantage of a surprise slip from previous Group B leader Sweden, which lost 2-0 in Georgia earlier on Friday (AEDT), to leapfrog Sweden into first place.
Spain had never lost away to Greece in four previous meetings, and a first-half penalty from Pablo Sarabia increased that record to four wins and one draw.
It will book its place at Qatar 2022 if it can avoid defeat to Sweden on Monday in Seville.
The visitor unsurprisingly dominated possession but created little early on against a well-organised Greece.
The home side thought it had taken the lead on 21 minutes as Giorgos Masouras finished well from a Thanasis Androutsos throughball, but the Olympiakos forward was offside.
Just three minutes later, Spain were awarded a penalty after Inigo Martinez was felled in the box by Dimitris Giannoulis following a corner, and Sarabia sent Odisseas Vlachodimos the wrong way to put La Roja ahead.
The host needed a win to keep alive any hopes of qualification itself and started to gamble a little more towards the end of the game, but Spain remained relatively calm as they eased to a vital win in Athens.