The change in tactics worked as United sprayed a barrage of long balls into Liverpool's penalty area in the final 10 minutes and finally scored the equaliser late on as Zlatan Ibrahimovic pounced on a Antonio Valencia cross.
"We were the better side. They play long balls in a wild game. We played the better football and had the better plan," Klopp told British media.
"In the end period of the game when United started playing long balls -- (to) Fellaini and Ibrahimovic -- after 80 minutes high intense football it is really hard," Klopp added.
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"Usually you can accept a draw at Manchester United but I think after the entire 98 minutes we could have deserved a win."
The draw put Liverpool third in the league, seven points behind Chelsea after 21 games, while sixth-placed United lost ground in the battle for the Champions League qualifying spots.
United next travel to Stoke City for a league game on Saturday, while Liverpool make the trip to face fourth-tier side Plymouth Argyle in an FA Cup third round replay on Wednesday.
(Reporting by Shravanth Vijayakumar in Bengaluru)
