Rangnick, 63, will be interim manager until the end of the season before taking up a consultancy role with the club for a further two years.
The 63-year-old German is due to start a two-year consultancy role after the season finishes.
It was Rashford's second goal in as many games here following a barren run of matches that saw him go 11 without scoring.
Ralf Rangnick and his City counterpart Pep Guardiola have both urged the Premier League to increase the number of substitutions from three to five.
Rangnick's deal is set to see him move into a consultancy role at Old Trafford after the end of the current campaign.
The German was appointed on Monday on a contract until the end of the season after the sacking of Ole Gunnar Solskjaer and will work in a consultancy role for a further two years.