Olid, 41, took over as manager in the summer, joining from Hearts, and promised to implement a new attacking style of play that focused heavily on possession.
However, in their opening two games, United had just two shots on target and less than 40% possession in both matches. In the second of those games, they suffered the five-goal humbling at home by Chelsea.
Few expected them to come away with anything against an Arsenal team who many tipped to win the title before the season kicked off, albeit they have struggled to score goals efficiently.
Olid knew it would be a test but said on Friday she wanted her players to believe in themselves, show confidence, and to go back to the defensive roots they knew and built so well under her predecessor Marc Skinner.
“It has been an intense week because there were a lot of emotions from players and staff. Nobody wants to lose 5-0,” said Olid.
“We grew as a team. The first day after losing 5-0 is not easy for anyone. We spoke and said we needed energy this week and a positive mindset.
“The focus was the defensive shape. It’s trying to get them more confidence.”
Olid ultimately wants her United team to dominate the ball and create more chances in front of goal, but this time they delivered on her plea to be defensively solid.
They looked well organised in an even first half in which United midfielder Simi Awujo stood out, while Andrea Medina tracked back effectively.
They won more duels than Arsenal, put in more tackles and completed more interceptions than their opponents – but still had only 40% of possession.
Former England midfielder Izzy Christiansen told Sky Sports: “Manchester United have done a terrific job defensively. Overall, it’s just been solid.
“It looks so much better. The addition of Awujo has added some running power and they look so much freer in the final third.”
Arsenal’s pressure grew in an edgy second half, but United held on for as long as they could and were unfortunate to concede with seconds left on the clock.
They had led from midway through the half, after Julia Zigiotti seized on a mistake from Georgia Stanway.
Smilla Holmberg’s late volley denied United the spoils, but Olid saw foundations she could build on.
“I think the attacking will come,” she said. “If you implement a new style of play, you need time.
“That effort, that intensity, can never be missed. We deserved three points for the hard work and the effort.
“Of course, we need to create more chances. That is why I’m here. We need time, but today is the minimum that we have to deliver. I hope we can improve the attacking side.”