
For Luton, who are in administration, a replay at Anfield is almost more valuable than progress, and the proceeds form that second tie will go a long way to aid the club in its battle for survival.
They were well worth the draw, and with more composure in front of the goal, might even have won it.
David Edwards had wasted a one-on-one and Drew Talbot had put a free header over for Luton when Liverpool, against the run of play, took a 73rd-minute lead.
Chris Coyne, Luton's captain, squandered possession on halfway, and when defender Chris Perry's subsequent clearing header was poor, substitute Andriy Voronin was left clean through.
His effort was saved by goalkeeper Dean Brill, but Peter Crouch followed in to put Liverpool in front.
It took the League One side just three minutes to level, though, as Riise, in attempting to prevent Talbot's cross reaching Edwards, hooked the ball into his own arm, from where it cannoned over the line.
Newcastle, whose manager Sam Allardyce remains under intense pressure, will go to a replay after being held to a goalless draw away to Championship side Stoke City.
Arsenal ran out two-goal winners at Burnley, but never had it easy despite the Championship club having Kyle Lafferty sent off for a studs up challenge midway through the second half.
Croatia striker Eduardo da Silva gave them a first-half lead and then teed up the second for Nicklas Bendtner.
Derby County had to come from 2-0 down to earn a replay against Championship side Sheffield Wednesday.
A dreadful error from goalkeeper Lewis Price gifted Wednesday the lead as he allowed a Mark Beevers volley to squirm through his grasp.
Then former Derby striker Marcus Tudgay capitalised on a weak back header from Darren Moore to make it 2-0.
But Derby hit back, Kenny Miller and Giles Barnes getting the goals in a seven-minute spell towards the end of the first half to make it 2-2.
Fulham twice had had to come from behind to earn a 2-2 draw against Bristol Rovers.
A deflected Danny Coles effort put the League One side ahead after three minutes, and, although David Healy levelled, Craig Hinton headed Rovers back in front.
But Danny Murphy crashed in a 25-yard drive to force a replay with 17 minutes remaining.