Downing leg injury blow for West Ham
Winger Downing was replaced by Joe Cole at half-time on Saturday after being caught by his former Liverpool team-mate Flanagan.
Kevin Nolan was shown a straight red card for a challenge on Jordan Henderson late on and manager Sam Allardyce had no arguments with that decision, but believes Flanagan was fortunate not to be punished for his first-half tackle that left Downing needing hospital treatment.
He told the club's official website: "I wouldn't disagree with it (Nolan's dismissal), but I would say that there was a very similar if not worse challenge from Flanagan in the first half on Stewart Downing.
"Henderson has not ended up in hospital but Stewart has and he needs 16 or 20 stitches in a massive gash across his calf and Achilles tendon.
"There is more danger in Flanagan's challenge than there was in Kevin's but you couldn't argue that he sent him off. What you do argue is that he has been sent off and Kevin hasn't."
Guy Demel's own-goal gave Liverpool the lead just before half-time and Mamadou Sakho doubled their advantage early in the second half.
Martin Skrtel's own goal gave West Ham hope, but the in-form Luis Suarez made it 3-1 and his shot was then deflected in off Joey O'Brien to compound the visitors' misery.
The London club have not won at Anfield for 50 years and are now level on points with third-bottom Crystal Palace.
Allardyce was left to rue West Ham's lack of fortune and inability to take their chances.
He added: "There was an element of luck that got Liverpool on their way. They weren't looking like they were going to score before half-time and, if anybody had the better chances before they scored, it was probably us.
"Unfortunately their keeper has made a couple of good saves and there were a couple of good blocks, then there was a ball through from Mo Diame to Modibo Maiga which just the slightest touch - he just cannot seem to pull it off at the moment, the lad."