The Spaniard had previously overseen eight matches without a victory, but watched on as his players battle back from an early 1-0 deficit to claim a 2-1 win at the Liberty Stadium.
But Mel, who took over from Steve Clarke in January, insisted that the three points were far more important than speculation surrounding his job security.
"My position is not important," he said. "What is most important is to win, for the fans and the players. The future for West Brom for me is good.
"This win was for the fans and for the players because the players this week worked very hard and the fans were very kind to me today.
"It was important we equalised early in the second half because confidence is a problem and it gave us the confidence to go on and win.
"Now we have 28 points and we have more confidence. We will be training with a smile on our faces next week."
Roland Lamah opened the scoring for Swansea in the second minute, before Stephane Sessegnon and Youssouf Mulumbu inspired a second-half turnaround.
And Mulumbu credited West Brom having emerged for the second period with renewed belief as a key factor.
"It is a good feeling when you score but I am more happy for the team because we needed the three points and we did it in the perfect way in the second half," he said.
"What made the difference? At half-time we spoke about belief. We know we are a good team even if every newspaper you read says we are going to go down.
"I think in the second half we showed the real face of West Brom and I hope we are going to keep doing that."
