David Moyes was left to rue a number of missed opportunities as his first home game as Sunderland manager ended in defeat.
The defeat leaves Moyes with two defeats from his first two Premier League games in charge of Sunderland after last week's 2-1 loss at Manchester City.
"I don't think we deserve to be 2-0 down at half-time," Moyes told Sky Sports.
"They scored a great goal, the boy [Stuani] had a great shot, and of course we could have closed things down quicker.
"We were disappointed with the defending for their second but overall I don't think we deserved to go in 2-0 down.
"I don't think we deserved to go in 1-0 down. But that's football, we didn't take our chances and when they go theirs they took them.
"We had good attacking opportunities today but we didn't quite take the chances."
Younes Kaboul's departure to Watford and Lamine Kone's back injury forced Moyes into naming an unfamiliar XI, featuring a makeshift back four, for the 143rd Wear-Tees derby.
The hosts' cause was hindered further when John O'Shea was forced off injured 10 minutes before half-time, but they improved after the break and halved the deficit through Van Aanholt.
Moyes admitted he was left with the "bare bones" and could not plug the gap left by O'Shea's departure, but he was keen to draw positives from the defeat, before setting his sights on bringing in new recruits to the club.
"I thought we played well, we had a great level of intensity but we ran out of it with 10 minutes to go having worked so hard," he added.
"We had so many boys near enough making their debut in the Premier League, so many young boys.
"We were bare bones today with the team we had, particularly at centre-half, not helped by losing O'Shea.
"We went out to try and win, the boys really tried to score in the second half and we were a bit unlucky we didn't get an equaliser.
"We're working hard, everyone knows we're short, no more than us here, but we'll get the players."