Coutinho threaded a superb pass through to allow Henderson to cancel out Christian Benteke's opener before Steven Gerrard converted a penalty to seal the points.
The 7.7 million euro January signing from Inter is only 20 but has quickly become a fixture in Brendan Rodgers' starting lineup, and Henderson was full of praise for his "outstanding" young teammate.
"It was a great start to the second half for us and it was a great ball from Philippe, and thankfully I managed to slot it in, so I'm over the moon," Henderson told the club's official website. "It was all about the ball from Philippe which was outstanding and I thank him for that one.
"Philippe is brilliant and I think everyone can see that. He's a real talent and he's young as well. He has been outstanding since he came into the team and hopefully he can keep that going."
Benteke put struggling Villa ahead in the first half but the Reds turned the game on its head after the break to earn the victory.
"We were really pleased with our second-half performance," Henderson stated. "I thought we came out and played some good stuff, defended very well, so overall we are delighted with the three points. The manager told us at halftime to up our game a little bit more. We were doing okay and probably were the better team in the first half, just, but in the second we took that on board, got forward more and got the two goals."
He added, "[Luis] Suarez did brilliantly to win the penalty for us and we all know Stevie is brilliant at penalties."
In addition to hitting the target himself, Gerrard preserved Liverpool's lead by brilliantly heading a Benteke effort off the line.
"It was unbelievable," Henderson said. "Stevie was outstanding again for us and he's been doing that for many years now so it's not a surprise to anyone."
The former Sunderland midfielder was in Rodgers' XI for the first time since the Europa League tie against Zenit St Petersburg last month and hoped he had taken his chance to impress.
"It's nice to play football," he admitted. "The team has been playing brilliantly of late. I haven't played as much as I would have liked but I got the chance here and thankfully we won. Hopefully we can win more games now."
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