A flare was thrown onto the pitch during the game and sectarian songs were chanted which resulted in clashes with riot police as fans attempted to break a cordon
Rovers defeated the Blues 3-1, winning the tie 7-2 on aggregate, and thus progressed to a semi-final date with Cork City.
However the game was marred by sectarian displays, with both sets of fans chanting provocative songs and a flare was thrown onto the pitch from a section of the crowd.
Fans of Shamrock Rovers then clashed with riot police when they tried to break through a cordon.
Eyewitness reports described the display of political banners and flags from both sets of supporters, as well as missiles being thrown from outside the ground.
Linfield manager David Jeffrey described the occasion as "a sad night for football", while local politicians condemned the violence on social media outlets such as Twitter.
"I'm disgusted that some Shamrock Rovers fans came to Belfast tonight to engage in sectarian chanting and violence. It has no place in sport," said Social Democratic and Labour Party (SDLP) MLA for South Belfast, Conall McDevitt.
"The behaviour of a number of Linfield fans in Dublin was also deplorable. The two clubs need to clamp down on those who are damaging the game."