For all their endeavour and enthusiasm going forward, the Lions lacked real quality in the final third and failed to capitalise on a spell of late pressure as their winless run at the Den stretched to six games.
For Charlton, it was their second stalemate in just a matter of days - they drew 0-0 against Huddersfield in midweek - offering new boss Riga hope the Addicks can climb clear of danger.
Millwall, who are anxiously looking over their shoulders at the drop zone, had the first chance inside 10 minutes as Simeon Jackson flicked the ball over to Steve Morison and, although his cross was initially cleared, Jackson crossed again only for an oncoming Fred Onyedinma to head wide.
On 15 minutes, Ed Upson was caught in possession and Bradley Pritchard sprung a counter-attack, slipping through Callum Harriott one-on-one but his shot went straight at the goalkeeper as Charlton pressed.
Soon after, Onyedinma found space in behind the midfield and laid the ball off to Jackson but his shot was fired into Ben Hamer's hands as the home side looked more likely to break the deadlock.
Millwall were continuing to dominate and a long ball forward saw Morison display great strength in the box to fend off the Addicks defenders, but his shot on the turn bounced just wide.
Five minutes before the break there was a shaky moment for Charlton as Hamer headed away Morison's through-ball into the path of Onyedinma, but Pritchard defused the danger and nicked the ball away.
A good phase of Charlton pressure was nearly rewarded just minutes before the break as a beautifully weighted pass from Diego Poyet left Millwall goalkeeper David Forde in no-man's land, but Pritchard was unable to make a solid connection.
Charlton nearly got off to a flyer in the second half as Rhoys Wiggins produced an early cross but, despite a good connection from Harriott, his volley flew wide.
Reza Ghoochannejhad nearly made an immediate impact after coming on as Harriott played him through but, after gliding through the box, his shot went straight at the keeper.
On 72 minutes, a corner from Millwall nearly brought the opening goal as Stefan Maierhofer was unmarked but his header sailed over the bar.
Eight minutes later, a surging run forward from Ghoochannejhad won his side a free-kick after a cynical challenge brought him down - but Jackson slipped at the set-piece and his strike hit the wall.
Owen Garvan's free-kick in the 89 minute saw Hamer at full stretch as he clawed the ball over his bar for a corner.
In stoppage time, Millwall had Charlton on the ropes as they continuously struggled to clear the ball but, despite their dominance, the Lions could not make it count.