A first-half penalty converted by Iliman Ndiaye was enough for the visitors to win 1-0 on Saturday.
Tarkowski told evertontv: “We got the goal and then we just defended well, and we knew the second half was going to be tough.
“I've watched (Brighton) against many teams. I think they came back against Man City here, so it shows how good they are. But we've been resilient, we've grinded it out with 10 men for 15 minutes or so and everyone played their part all over the pitch from the front to the back.
“It’s been a tough period. So, to go back-to-back (with wins), coming here, a real tough place to come and to be down to 10 for whatever it was at the end and just grind it out.
“It feels big, feels really big, but we don't get carried away, obviously, because there are plenty of games to go. But to win back-to-back like that in a week feels really good.
“So, I’m really proud of the team there and let's move on now and go again.”