Elanga, 19, scored for the second game in a row while Andreas Pereira was also on the scoresheet with an outrageous volley, but Shandon Baptiste and Bryan Mbeumo twice equalised for Brentford as they prepare for their first season in the Premier League.
Despite waiting for a number of his international players to return to training, Solskjaer was able to pick a strong team against Premier League new boys Brentford including Tom Heaton, Aaron Wan-Bissaka, Nemanja Matic, Juan Mata, Jesse Lingard and Mason Greenwood.
Elanga also started after his goal during the 4-2 defeat to QPR on Saturday and followed up with another here after 12 minutes.
Wan-Bissaka crossed from the right and the Sweden Under-21 international, who scored his first competitive goal against Wolves on the final day of last season, volleyed his finish into the corner despite falling backwards.
Solskjaer said after the game at Loftus Road a decision on Elanga's immediate future is still to be made and he did his hopes of staying with the first team squad next season no harm at all with another impressive display.
Baptiste equalised for Brentford before half-time when his shot from the edge of the box dipped into the top corner but the quality of the strike was soon out done by Pereira.
Shortly after the break, Ethan Pinnock's header clear dropped to the midfielder 30 yards out but instead of bringing the ball down, the Brazilian rattled in a volley off the underside of the crossbar.
It looked to have won United the game until Brentford substitute Mbeumo equalised 10 minutes from time. United are next in action against Championship side Preston on Saturday.