Shola Ameobi and substitute Paul Dummett were both sent off in the second half as Liverpool came from behind to take all three points thanks to goals from Daniel Agger and Daniel Sturridge to cancel out Martin Skrtel's unfortunate own goal.
Ameobi was sent off for dissent for picking up two quick-fire yellow cards before the restart of the game after Liverpool's equaliser.
Pardew defended Ameobi's character and feels Dowd failed to "manage" the game properly as Newcastle ended the season in disarray.
"I know Shola. He doesn't swear. I told the ref he doesn't swear so how angry can he possibly get," said Pardew
"I don't actually know how angry you can get. He said: 'If you carry on I am going to send you off'.
"Whether it was sarcastically I don't know but Shola said: 'You are going to send me off!' And he did.
"That's sort of explanation that Phil gave me. So I think he should have managed that far better than that.
"It was shame. Liverpool then got two goals but with 10 men it was always going to be difficult."
Pardew felt the defeat was harsh on his side, feeling Dowd's performance played a huge role in the defeat.
"I thought our season in terms of performance was strong," added Pardew.
"We had a good game plan. Perhaps we were slightly unfortunate not to get the second goal when Gouff (Yoan Gouffran) was one on one.
"The ref played a part in my view in the second half.
"He should have managed the game a lot better than he did. And we conceded two goals.
"He had a part in the first one by Vurnon (Anita). Then there was a foul from Agger on Shola.
"Liverpool have had a fantastic season and they showed their quality. If you give too many set plays away it will hurt you.
"Unfortunately we didn't quite do as well as we should have."