Dawson, along with Australian winger Mark Bartholomeusz and Ireland 'A' prop Tom Court, made their home debuts in a game that hung in the balance until Maxwell's second try in the last minute.
Tim Barker, who has returned to Ulster after a two-year stint with Scottish side Glasgow Warriors, partnered former Wallaby Justin Harrison in the second row against the Titans.
Loosehead prop Justin Fitzpatrick tries to shrug off the challenge of Titans lock Erik Lund at Ravenhill.
New Zealander Paul Steinmetz, who formed a midfield partnership with Paddy Wallace on Friday, gains the hard yards for Ulster. Steinmetz ended last season on the sidelines due to a fractured cheekbone.
Ulster full-back Bryn Cunningham on the attack against the English National One side - Ulster claimed an overall victory (65-21) in the Grafton Challenge with the province's 'A' team also beating Rotherham 'A' 39-7 on Saturday. Ireland Schools-capped flanker William Faloon scored two tries in the final 10 minutes to cap off the win.
Rotherham coach Andre Bester, who was formerly in charge of Belfast Harlequins, was still pleased with the trip across the water. Praising the Ulster defence, he said: "We played against the best defence in European rugby on Friday. It was a good experience and it will prepare us for grinding out results on the road this season."
**All photos by Andrew Paton of Inpho Photography**