The event is described in the 2009 book The Guards Brigade in the Crimea by Michael Springman.
On pp 81-82 the encounter is described thus:
Captain Goodlake and Sergeant Ashton advanced along the ravine to guard against an ambush and to examine the caves. While they were inside a cave, a party of Russians, 600–800 strong, became visible to the sharpshooters who fired on them and retired along the ravine. Goodlake and Sergeant Ashton burst out of the cave and fought their way down the slope. Owing to their grey overcoats and flat caps, they were able to lose themselves in the mass of grey-coated Russians who were advancing along the bed of the ravine.
The author quotes elsewhere from Goodlake's personal correspondence, which may have been the source of this story.
While not explicitly mentioning their great coats, the incident is described in a paper titled Origins of the Regiment about the history of the Coldstream Guards.
... Goodlake and Sergeant Ashton went ahead to the caves and left the rest of the patrol to keep watch. Unfortunately a large Russian column marched past the caves while they were inside, and they were heading towards the sharpshooter group. When the group saw the Russians coming towards them, they assumed that their officer and sergeant had been caught. They opened fire on them and began to withdraw to a ditch from where they could hold off the superior force.
Goodlake and Ashton came out of the cave and ran in amongst the Russians, pushing their way to the front where the two sides were exchanging fire. Goodlake shouted out to his men to let them know they were there and he and the sergeant dashed over to them. They held off the Russians until relieved by 2nd Bn Rifle Brigade.
So although their great coats are not mentioned, the fact that they "ran in amongst the Russians" does suggest that their attire may have caused some confusion.
In fact, Goodlake did make a passing reference to the incident in a letter to his parents dated 6 November 1854. The letter is included in Sharpshooter in the Crimea: The Letters of the Captain Gerald Goodlake VC 1854-56, Volume 3 by Michael Springman:
... I and a Sergeant were nearly caught in a cave but we made a bolt for it and got off with only a bullet through my coat and he shot in the arm.
This was the action for which Goodman was awarded the Victoria Cross (hence the link to the London Gazette from the Wikipedia page).
Lastly, it's worth mentioning that the incident was reported - with varying degrees of detail - in a number of contemporary British newspapers. One example was the Illustrated London News of Saturday 20 January 1855:
... and were only obliged to retreat by sortie from a breastwork in their rear. Captain Goodlake had a very narrow escape, he was left with a sergeant, and had to cut his way through them, luckily with no more damage than a bullet through his coat"
In this case, they cite as their source "A private letter from the camp"