Elanga was a regular for the Red Devils last term, making 27 appearances in all competitions, contributing three goals and three assists in the process.
Laird, meanwhile, spent time on loan at Swansea City and Bournemouth last season, and the 20-year-old has only made two first-team appearances for Man United during his time at the club.
Elanga is expected to be an important player for the Red Devils next term, but it is unclear whether Laird will remain at Old Trafford or head out on another loan spell.
Both players featured in Friday's pre-season friendly with Melbourne Victory, playing 45 minutes each, and Elanga has said that he was "really happy" with the outcome, as Ten Hag's side won 4-1.
"It was a great crowd," Elanga told MUTV. "From the warm-up, we could tell that they were on it. We conceded early on and it's was important to get the two goals and settle the game. In the second half [after 10 substitutions], the second team did amazing and controlled the game. I'm really happy."
"It's healthy competition It's important to have competition like that in training because, if someone starts ahead of you in games, then it will motivate you to work even harder in training to try to take their position in a game. It's always important to have competition in the team."
Laird, meanwhile, added: "It was good.It's always nice to be on the pitch as we're footballers and we love playing football. We keep building, keep building our fitness and it's a good result. I'm happy with it.
"The main thing was the time difference and plane journeys, jet-lag, just getting yourself ready again. That's part of our preparations and that's what we have to do as professionals. It's something we have to deal with and we prepared well, we delivered well on the stuff we worked on. I think we were solid today."
The duo will now be hoping to feature in Man United's third pre-season friendly of the summer against Crystal Palace on Tuesday, which comes ahead of their final game in Australia against Aston Villa next weekend.
Diogo Dalot is expected to be the team's first-choice right-back next season, with Ten Hag allegedly making Aaron Wan-Bissaka surplus to requirements at the club.
Should Wan-Bissaka depart, Laird could be promoted to second-choice right-back at Old Trafford, with the club's academy product set to be involved in the club's entire pre-season campaign.
Charlie Savage and Zidane Iqbal are also among the youngsters to impress in Man United's first two matches against Liverpool and Melbourne Victory.
The Red Devils will play their final two friendlies of the summer against Atletico Madrid and Rayo Vallecano on July 30 and July 31 respectively before opening their 2022-23 Premier League campaign against Brighton & Hove Albion on August 7.
Man United confirmed Christian Eriksen as their second signing of the summer on Friday, while Lisandro Martinez is believed to be on the verge of joining the club from Ajax.