Virgil van Dijk's header on the hour mark was enough for the Wednesday night's victory.
Slot said afterwards: "It didn't feel to me at all as a struggle because I think we played really, really, really well again. We didn't face the weather, we faced Sunderland, who were until tonight unbeaten at home.
"They are a good team, (have a) good manager and (have a) difficult playing style to play against because, as we know from the game we played at home against them, it's a lot of long balls you have to defend and a lot of balls into the channels. I think we did that really well.
"We controlled the second ball really good and from that second ball we played some very good football. What I'm getting used to more and more now is that we miss a lot of chances, which we did tonight again.
"But the big exception for us tonight compared to all the other games we played against promoted teams, or teams that we were the better team, we were not able to score from a set-piece. Usually we concede one, but tonight we could score and that's the difference for us in this game."
The win followed Sunday's disappointing home defeat to Manchester City.
Slot also said, "I don't think that's a surprise to anyone if you follow this season. We've had so many setbacks, so many disappointments, so many times that we were the better team on the pitch and we didn't get what we deserved. But three days later the players performed again, showed up again and play usually a very good game.
"Not always leading to a result but always leading to the fact that we were the better team on the pitch. As we were, in my opinion, tonight. After only two days off, an away game in difficult circumstances because it's not only raining but the pitch was of course heavy as well, usually that can impact something if you had one day less to recover compared to the other team, but I don't think I saw that today. I think I saw a very good performance from our side and throughout the 97 minutes."