Teenage winger Zaha, who scored twice against Crawley in the opening round and set up both goals as Wigan were sent packing last week, put the Eagles on their way early in the first half at Selhurst Park on Tuesday night.
And it was from Zaha's cross that Andrew, who last scored for Palace 17 months ago, marked a rare opportunity in the first team with the second after the break.
Merouane Zemmama pulled one back for the visitors but Palace held on to earn themselves a touch of revenge after Boro beat them 1-0 in the Championship on Saturday.
The fact there were only seven survivors from the teams which lined up at Selhurst Park three days ago illustrated where the priorities of both clubs lie.
With Boro's top scorer and weekend matchwinner Marvin Emnes starting on the bench and Palace pinning their hopes on Andrew, who had previously scored three times in three years with the club, hopes of a feast of goals were not exactly high.
But it took just 19 minutes for Palace to take the lead when Jonathan Williams skipped past Kevin Thomson in midfield and sent Zaha scampering clear to lash the ball past Danny Coyne and into the net.
With Palace in the ascendancy throughout the first half, Andrew forced Coyne into an unconvincing save with a 25-yard drive and also had a header deflected wide just before the break.
Glenn Murray came agonisingly close to finding the net after the break when he met Dean Moxey's cross with a deft volley which bounced on top of the crossbar.
Coyne then twice denied Murray from long range, but the second goal did come after 52 minutes when Zaha's deep cross found Andrew at the far post.
The former Luton striker planted his header back over Coyne and although Boro defender Stephen McManus helped it over the line, no one was going to tell Andrew the goal was not his.
Boro were thrown a lifeline three minutes later when Zemmama's shot flew across Lewis Price and into the net.
But Price came to Palace's rescue when he tipped Alex Nimely's effort wide to ensure the hosts' cup adventure goes on.