Kilmarnock took the victory, but it was Queen of the South who will grab most of the plaudits after coming close to forcing a Scottish Cup replay despite playing nearly 90 minutes with 10 men.
Stephen McKenna was sent off for a rash early challenge and Cillian Sheridan fired Killie ahead on eight minutes.
Borja pounced on Sheridan's cross to extend the lead in the second half.
Nicky Clark headed a reply eight minutes later, but it was not enough for the Second Division side.
And it is their Scottish Premier League opponents who march on slightly fortuitously to the fifth round after a cup thriller at Rugby Park.
Those looking for a Scottish Cup fairytale had their eyes set on Ayrshire, with the visitors managed by former Kilmarnock hero Allan Johnston, riding high in Division Two, having accounted for Hibernian and Rangers in knock-out competitions this season and being unbeaten in 10 away games this season.
Scottish Cup history also favoured Queens, who had lost only once in nine meetings between the sides in the tournament, albeit their last encounter came 23 years ago.
However, it was their most recent meeting that provided the most telling omen.
Last September, Kevin Smith's early dismissal was held responsible for the comprehensive nature of the Dumfries side's League Cup exit.
This time, it was McKenna's turn to be the villain with a rush of blood to the head and his late challenge led to Borja requiring treatment and Mike Tumilty waving what many considered a harsh red card.
Queens goalkeeper Lee Robinson had already had to prove his reflexes with a fine double save by the time Killie made what looked like an inevitable breakthrough.
Sheridan met Borja's corner at the back post to continue his scoring streak since arriving at Rugby Park.
However, Kilmarnock, minus the talismanic Manuel Pascali through suspension, laboured to take advantage of their extra man and Queens, wearing a black and white strip more associated with Killie's local rivals, Ayr United, ought to have been level by half time.
It was perhaps no surprise that former Ayr defender Chris Mitchell looked particularly inspired by the occasion and tested goalkeeper Cammy Bell with a low free-kick before screwing wide another effort from the edge of the penalty box.
That latter move had come from a fine Clark break and the in-form striker also sent a looping drive over the top.
Queen of the South were fortunate not to go further behind when Gary Harkins' dipping 20-yard drive struck the face of the crossbar.
However, Killie were just as fortunate that Clark could only bundle the ball wide from six yards after Paul Burns had pounced on some slack home defending.
Killie came out with more purpose after the break and, during the early siege on the visitors' penalty area, Sheridan sent a header flashing across the face of goal and a powerful drive against the crossbar.
Yet Queens almost snatched an equaliser on the break, with Bell showing his international class with a point-blank stop to deny Derek Lyle.
Bell was the Killie hero again and his save from close range from Ryan McGuffie led to the second goal that virtually killed the tie.
The home side immediately broke to the other end, where Sheridan's low cross was poked into the corner of the net off Borja's outstretched toe.
Queens just would not accept defeat and Clark headed home from five yards following a McGuffie cross.
However, while they must take credit from avoiding a repeat of the 5-0 thrashing they suffered at Rugby Park last season, they had given themselves too much of a mountain to climb and the 10 men came up just short.
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