First-half goals from Patrick Agyemang and David Connolly had put the visitors in the driving seat at Gresty Road before Mathias Pogba pulled one back for Crewe.
But despite dominating second-half possession, Crewe could not find the equaliser and the south-coast club held on to complete the double over their hosts in League One.
Whittingham said: "There is a huge sense of relief. We didn't like to say it,but we felt it was building and the team was starting to gel so to come away and get that first win in many a while is just great.
"You can't be happier for anyone than the fans who have been superb again - they haven't stopped singing and this is as much for them as for the players.
"We were getting worried and tense towards the end which is natural for a team that hasn't won for so long. It was nerve-racking, but we got our bodies on the line and held on.
"It was like we had got promoted when the fans came on the field at the end, but we have to keep things in perspective.
"It's one win and there is another game next week now. Hopefully this will give us some confidence to move on and get some more points on the board.
"Our first-half performance has got to rank among our best for a while and it's good to see the players buying into what we are doing on the training field.
"There are a lot of people working hard at the club and this is one for the whole club and anyone who has dedicated themselves to helping the club."