A third-round upset had always appeared unlikely at the Memorial Stadium, with the League Two side in disarray following the midweek departure of manager Paul Buckle and a run of form that had seen them fail to record a league win since October and fall to 19th in the table.
Buckle's assistant Shaun North was placed in caretaker charge and, while his side battled gamely, a famous victory never looked likely despite veteran forward Scott McGleish, who turns 38 next month, pulling one back in stoppage time before missing a penalty to reduce the deficit further.
By that point the Premier League outfit were already assured of a place in the fourth round thanks to goals from Marc Albrighton, Gabriel Agbonlahor and a lovely solo effort from Ciaran Clark.
And McLeish, who kept his promise to field a strong side, was delighted with his players' application.
"We treated this game with great respect, it is a fantastic tournament and we felt the players had enough rest between now and next week to play," he said.
"I thought Marc Albrighton helped us up the tempo of the game as we wanted to and we dealt with any forays they had going forward, we were extremely professional in that respect and we did not give them a look in until late in the first half.
"The pitch was a bit wobbly at times and our delivery into the box was not as good as we would have wanted but that is understandable.
"The players handled it very well, the TV cameras were here tonight because they thought there might be an upset but we dealt with that. We were a bit complacent for their goal and the penalty but overall I am pleased."
However, the victory did come at a cost as Emile Heskey was forced off at half-time with a recurrence of his Achilles problem, meaning he faces three to four weeks on the sidelines.
Heskey's absence means completing a short-term loan deal for Keane becomes a greater priority for McLeish.
But he is confident that the 31-year-old Republic of Ireland captain will link up with the Villans over the next 24 hours despite "red tape" holding up the deal.
The Scot came close to signing Keane when he was in charge at Birmingham 12 months ago but the Irishman chose to make the move to West Ham instead, where he made 10 appearances and scored just two goals as the Hammers were relegated from the Premier League.
But McLeish believes Keane is in better shape physically than he was a year ago and that the motivation to be at his best when leading his country in Poland and Ukraine in the summer, as well as keep sharp before the MLS resumes in March, is a major positive.
He said: "We have had a report that he looks in much better shape then he did at this time last year when he went to West Ham, he is in tip-top shape.
"The fact that he has the tournament coming up and he wants to be at his peak, and if he feels this is the right way to get his sharpness and he has that kind of motivation then it can only be good for us.
"He has enthusiasm, he brings leadership qualities as well that can help any dressing room and he scores goals, has great movement and hurts defenders. I know all about his pedigree.
"I have worked with some top players in forward positions like Michael Mols and Ronald De Boer, and we have some great players here and Robbie will add to that."
For Rovers it is now about securing their Football League status for another season.
The likes of Sean O'Driscoll and Gary Johnson have been linked with the vacancy, but North says he has not been told of an impending appointment.
He said: "I have been told nothing more at the moment (about a new manager). I will start preparing for Tuesday's game against Hereford unless I am told otherwise.
"I imagine the board will let me know what is happening as soon as possible because any newcomer who takes over will need time to prepare the side in his own way."