The Steelmen had arrested a five-game winless run which included a 6-1 home defeat to Rangers before the international break, with a 2-0 win at Aberdeen.
And it was a seamless return to action at home where attacker Connor Shields scored his first goal since signing from Queen of the South in the summer with a close-range finish after 23 minutes for the interval lead.
Undoubtedly the turning point came in the 65th minute when the lacklustre Tynecastle side went down to 10 men and immediately lost a second.
From the free-kick which saw Taylor Moore sent off by referee Willie Collum for picking up a second yellow card, defender Rickie Lamie headed in a second to keep the Steelmen in fifth place ahead of the trip to Dundee next Saturday.
Alexander said: "We need to focus on the next one. Every week we are tested to the hilt by every opponent.
"It is a real tough division, we have seen that already, we have played everybody.
"But we also know we can give people a real tough game and we have shown that in the last two games.
"We have to pick the bones out of Saturday, we have to analyse it, we have to understand what we did really well. There is still room for improvement, I feel.
"But we will continue to work on the training pitch and I know my players, because I have seen a them for a good few months, will always commit to their work and do their best and they are getting their rewards."
Hearts boss Robbie Neilson was left unimpressed by the whole afternoon.
He said: "It was a big game for Motherwell, you could see by the way they were celebrating at the end.
"We have to handle it. We know it is going to be direct, second balls, free-kicks, you give them the lead they are going to kill the game so you live and learn."