Luke Williams gave 'Boro the lead going into half time and two second half goals from Grant Leadbitter and Kike see Middlesbrough progress.
It was the Championship side that settled into the game quicker, passing the ball neatly and showing attacking intent.
Defender James Wilson received the first booking of the night after catching Leadbitter as he raced onto a loose ball.
The visitors were enjoying the better of the possession but did not really trouble goalkeeper, Paul Rachubka.
Leadbitter, who was a constant threat throughout, attempted to dink Latics' goalkeeper just before the half hour mark, but the ball fell just wide of Rachubka's goal.
'Boro did take the lead on 30 minutes through a wonder strike from Williams. The striker found plenty of space in the middle and struck from 25 yards and the ball thundered past the goalkeeper and into the top corner.
Latics did start to come back into the game but were struggling to create any real goal scoring opportunities.
The home side made two changes at the break with Winchester and Dayton being brought into the game replacing David Mellor and William Gros.
The hosts got off to the worst possible start in the second half as the instrumental Leadbitter doubled Middlesbrough's lead.
A quick throw in was not dealt with by the Oldham Athletic defence and the ball cut back to the midfielder who confidently stroked home from 12 yards on 48 minutes.
With the rain pouring, Latics' hopes of reaching the second round of the competition dampened as they fell three goals behind.
Substitute Kike's classy turn of pace and finish was out of Rachubka's reach and the Spaniard made it two goals from two in a 'Boro shirt.
The goals started to dry up as both sides struggled with the greasy conditions.
The visitors showed their Championship status, confidently keeping possesion with Latics struggling to get retrieve the ball.
With time running out, Timothee Dieng unleashed a volley in stoppage time but the ball deflected wide.
Seconds later, the referee blew the final whistle and it is Middlesbrough that progress to the next round of the competition.
Oldham Athletic: Rachubka, Mills (Brown 74'), Dieng, Elokobi, Kelly, Philliskirk, Mellor (Winchester 45'), Clarke-Harris, J.Wilson, B.Wilson, Gros (Dayton 45').
Subs not used: Coleman, Kusunga, Bove, Forte.
Middlesbrough: Konstantopoulous, Woodgate, Leadbitter, Nsue, Hines, Reach (Kike 57'), Ayala, Williams, Fewster (Adomah 74'), Husband, Whitehead (Smallwood 70').
Subs not used: Mejias, Omeruo, Bennett, Tomlin.
Goals: Oldham Athletic 0-3 Middlesbrough (Williams 30', Leadbitter 48', Kike 60')