Manchester United manager Jose Mourinho applauded Liverpool's defensive display, while opposite number Jurgen Klopp said the hosts resorted to long balls in a feisty draw at Old Trafford.
United striker Zlatan Ibrahimovic's late header levelled James Milner's penalty on Sunday, leaving the Reds without a win in four matches.
"We attacked and Liverpool were the team that defended," said Mourinho.
Klopp said: "We were the better side. They play long balls in a wild game."
"We played the better football and had the better plan," added the German.
Mourinho was criticised for negative tactics in United's goalless draw at Anfield earlier in the season but said he wanted to "see if the critics are fair" after Sunday's draw, after which he said Liverpool were "happy with a point".
"They were clever," added Mourinho. "They took their time, they know how to play football and control the emotions of the game.
"They knew they would be in trouble in the final few minutes."
Defensive Reds a pain in the neck for Mourinho
Mourinho said the number of men defending Liverpool's area prompted his decision to introduce the physical Marouane Fellaini for full-back Matteo Darmian on 76 minutes.
The Belgian headed against the post in the build-up to Ibrahimovic's equaliser, although the switch to a more direct style triggered by the Belgian's arrival resulted in just five United touches in the Liverpool area.
Mourinho added: "The people need to know what Marouane Fellaini is great at and what he is not so good at. Marouane is very good in some aspects."
He added: "I have a problem with my neck because I was always looking to the left in the second half and I saw so many yellow shirts in front of me I thought 'let's go for it'.
"We lost two points when we wanted all three."
United, now unbeaten in the league since October, remain sixth, two points adrift of local rivals Manchester City and 12 behind leaders Chelsea.
Klopp strongly refuted suggestions the draw was key in the title race, with his side now third, seven points behind Chelsea.
Liverpool were without Sadio Mane - who scored for Senegal at the Africa Cup of Nations - and defender Joel Matip, who was not given international clearance after turning down the chance to play for Cameroon in the tournament.
Klopp added: "When you see the line-ups and our circumstances, we had lots of little issues, you say 'a point at Old Trafford - let's take it and go home.' Of course it now doesn't feel like that because of the performance of my boys."
Klopp believes Liverpool were "dominating" until the 75th minute and though his side only had 45% of possession on the day, they did better the shot count with 13 to United's nine.
The Reds ran 12km further than their hosts but are now on their longest winless run of the season in all competitions.
"In the end period of the game when United started playing long balls - [to] Marouane Fellaini and Zlatan Ibrahimovic - after 80 minutes high intense football it is really hard," added Klopp.
"Usually you can accept a draw at Manchester United but I think after the entire 98 minutes we could have deserved a win."