Hagi, who had been absent since testing positive for Covid-19 while on World Cup qualifying duty with Romania earlier this month, returned as a second half substitute against Livingston at Ibrox.
With the tie goalless after a scrappy first half, Hagi lit up proceedings as goals from Kemar Roofe and Alfredo Morelos saw Rangers progress.
“We are delighted with the result,” said McAllister. “It is a case of job done. I thought in the first half we huffed and puffed, but we did create a few chances.
“Again, we felt, speaking to the players at half-time, we were wasteful in the final third. We were lacking a wee bit of quality and a wee bit of guile to create that quality pass and get that strike on goal.
“We made a change at half-time, the manager did, and it paid dividends. I thought Ianis brought that quality and guile we were missing.
“Credit to him, because he’s had a tough time recently with Covid and ten days in quarantine.”
Hagi expressed his satisfaction with the outcome and admits Rangers are finding it tough to match the high performances levels they produced last season.
“As long as we score goals and win I think everyone is happy,” said Hagi. “Obviously what we achieved last season, we set the standards so high. It will be really difficult for any team in the future to match it.
“But we are Rangers and we have to go again every single year. We are happy we have so many games to prove ourselves again and again.
“It feels good tonight. We had to do the job. We won the game 2-0 and now we concentrate on the next few games. They come thick and fast.
“The manager just told me to go out and do my job as a number ten and bring some quality to the team. I had to create goals, assist goals, just do my job and I am happy I did it.”