Town are leading Fulham 1-0 via a Daryl Murphy goal at half-time.
Mick McCarthy handed Hungarian frontman Bajner his debut alongside Murphy, while Elliott Hewitt was at right midfield with Jay Tabb on the left and Luke Hyam and Cole Skuse in the centre.
At the back, skipper Luke Chambers retained his right-back position with Tyrone Mings at left-back and Tommy Smith and Christophe Berra at the centre of the backline. Dean Gerken was in goal.
McCarthy named no defenders on a bench which included strikers David McGoldrick, Frank Nouble and Jack Marriott.
Fulham included eight debutants in their starting line-up, including £11 million striker Ross McCormack.
Despite the Cottagers enjoying most of the early possession, the Blues were first to threaten in the fifth minute. A freekick on the left was played short to Mings, who whipped over a cross but Fulham debutant keeper Jesse Joronen superbly tipped over.
After the resultant corner had been cleared, Skuse left the ball to Hyam, who hit a low shot across the face of goal and wide.
Chris David wasted a good opportunity for the visitors in the 11th minute when he scraped well wide of the post from the edge of the box after Tim Hoogland had broken strongly from the back.
The Blues went close again in the 17th minute when Hewitt aimed a long throw from the right towards Bajner. The ball defected out to Skuse on the edge of the box and the former Bristol City man smashed a shot only just over Joronen’s bar.
There was a big scare for the Blues in the 25th minute when Gerken spilled David’s 20-yard shot. But, having made the initial error, the keeper reacted brilliantly to block Moussa Dembele’s close-range rebound.
Fulham continued to enjoy significantly more of the ball but just after the half hour Skuse again went close, this time with a looping 25-yard effort which went a couple of yards beyond Joronen’s right post.
And two minutes later the Blues went in front. Hyam won the ball and burst through the middle before finding Murphy, who left Whites centre-half Shaun Hutchinson standing as he took the ball into the area and hit a right-foot strike which beat Joronen.
Soon after the restart goalscorer Murphy curled another effort wide after good work Mings. On 37 Hyam got his name in referee Stephen Martin’s book for a late tackle on Dembele.
As the half moved into its final five minutes Town continued to threaten, the previously quiet Bajner winning a couple of Hewitt’s dangerous long throws into the area.
On 42 Emerson Hyndman misjudged a long ball down the middle, causing Hutchinson to make a desperate clearance which cannoned off Bajner which, fortunately for the home side, flew through to Joronen.
Two minutes later, after David had shot over for the Cottagers, Bajner was replaced by McGoldrick, returning to competitive action for the first time since suffering a knee injury in February.
Fulham will have counted themselves unfortunate to have gone in behind after a half which they had controlled for the most part.
However, aside from Dembele’s opportunity handed to them by Gerken’s error, the visitors hadn’t created too many chances.
Prior to Murphy’s well-taken goal, only an excellent save from Joronen had prevented Berra from opening the scoring, while Skuse and Hyam might have done better with efforts from the edge of the box.
Town: Gerken, Chambers (c), Mings, Berra, Smith, Skuse, Hyam, Tabb, Hewitt, Bajner (McGoldrick 44), Murphy. Subs: Bialkowski, Bru, Wordsworth, Henshall, Marriott, Nouble.
Fulham: Joronen, Hoogland, Stafylidis, Hutchinson, Bodurov, Parker (c), Dembele, Hyndman, David, Burgess, McCormack. Subs: Bettinelli, Eisfeld, Roberts, Woodrow, Rodallega. Referee: Stephen Martin (Staffordshire).