Eddie Howe's side went agonisingly close to opening the scoring in the first-half as Joelinton struck the bar twice in succession, with the Brazilian's looping header hitting the crossbar before his rebound clipped the left-hand post.
Cristiano Ronaldo saw two goals disallowed in the second-half for the hosts before Newcastle withheld a late attacking surge from the hosts to record a fourth Premier League clean sheet of the 2022/23 campaign and currently lie four points adrift of the division's top four.
Botman, who joined Newcastle from Ligue 1 outfit Lille last summer, told nufc.co.uk: "It was a hard day for all of us. Coming here is difficult as they're a good team with a lot of quality but I think we are in a good shape at the moment. We're really sticking together and that's what we wanted from today.
"Of course, we hoped for more. We always want to win, no matter who we play, but I think we have to be happy with a point. It was a good team against a good team and we now move onto Wednesday (against Everton).
"Coming here and getting a clean sheet is always good. I think every opponent in this competition is very tough and there are lots of good strikers in this league so it's a good point.
"This league has very good strikers against you and we have to do well as a team. It starts from the front and we were very solid. We always hope to get the win but getting a clean sheet is very pleasing for us."
The Magpies extended their unbeaten streak to five matches following a commendable point against Erik ten Hag's side and quickly return to league action on Wednesday night when they host Everton at St. James' Park (kick-off 7.30pm BST).
Botman, who began his career at Dutch giants Ajax, is determined for United to extend their run further ahead of facing Frank Lampard's Toffees under the lights on home soil.
The 22-year-old centre-back added: "For every team, it's always nice to get an unbeaten streak and we want to keep it going for as long as possible. We want to get more wins instead of draws but I think we have to be realistic and, today, a draw was good for the team.
"It'll be my first night game at St. James' Park but it's always a pleasure to play at that stadium and in front of the amazing fans. It will be great to see them again on Wednesday!"
