Magpies skipper Jamaal Lascelles - who had earlier produced an exceptional goal-line clearance to deny Tammy Abraham - headed the victor from Matt Ritchie's 76th-minute corner to end Newcastle's run of 12 matches without a Premier League away win.
For Benítez, the outcome was just what the doctor ordered.
"I'm really not satisfied with the way we played". We had good training week, all our meetings were very good.
"I'm finding it hard to understand why I saw what I saw". We got the overload in midfield but we never got control of the game. "Even right up to the end, where we did continue to push, we couldn't get a decent ball into the box", he added.
After securing their first win of the campaign with a 3-0 home victory over West Ham before September's global break, the 31-year-old now believes that Newcastle can get their season off properly - starting with another possible three points against the Swans.
Matt Ritchie was lucky not to be shown a red card early in the second-half
Swansea should have taken the lead on the hour mark when Leroy Fer released Abraham, and the striker rounded Elliot for what looked a simple finish.
They were fortunate to finish with 11 men after Matt Ritchie escaped a red card for the sort of high foot just after halftime on Alfie Mawson which saw Sadio Mane sent off for Liverpool at Manchester City on Saturday. "He said we are now under way".
Deadline-day signings Renato Sanches and Wilfried Bony are set to make their Swansea debuts at home to Newcastle on Sunday after a busy summer of transfer activity at the Liberty Stadium.
"I'm very happy with the way that the squad has shaped up", head coach Clement said.
"He has been active in the game and kept in touch with us throughout the game". We need Rafa back as soon as possible. "But, at the same time, we've got to wake up quickly".