The Dutchman followed up his first-half brace with a second-half penalty to take his tally to four already this season after he also netted on the opening day of the league season.
Boro's progression extends their sequence of not having exited in the first round of the League Cup since 1985, but for League One Walsall it now represents their worst ever run in the competition having been eliminated at the first hurdle for a fifth successive season.
Boro boss Tony Mowbray made five changes to the XI who drew 2-2 with Portsmouth on Saturday - Danny Coyne starting between the posts to make his 500th career appearance.
But it was Saddlers counterpart Dean Smith who was forced into an early reshuffle after just 12 minutes when midfielder Adam Chambers, match-winner on the opening day against former club Leyton Orient, was dismissed for a two-footed challenge on Richard Smallwood.
And it took Boro little time to capitalise as Emnes fired a low effort into the bottom corner from 18 yards after the hosts had failed to deal with a ball from the right.
Emnes doubled both his and Boro's advantage after 37 minutes, converting Justin Hoyte's left-wing cross in style.
And the tie was over as a contest seven minutes after the break as Smallwood was brought down in the area by Kevan Hurst, leaving Emnes to convert his third from the spot and walk home with the match ball.