Both of Ipswich's new loanee strikers were involved in the opening goal as David Healy claimed the ball on the right wing from returning skipper Jon Walters and his deflected cross was touched past Danny Coyne by Murphy who had got there just before Wheater.
Middlesbrough were powder puff in the first half, despite fielding new signing Lee Miller, although Town themselves had few clear-cut chances even with the majority of the possession.
Coyne saved a 25-yarder from Walters before the visitors finally showed a bit of attacking intent when skipper Gary O'Neil strode forward to the edge of the box, but his shot was blocked by Damien Delaney.
Murphy had a chance for a second after pulling down a David Norris pass but his miscued volley was easily held by Coyne.
Boro finally forced Arran Lee-Barrett into a save on the half-hour, but it was an easy one from a tamely struck 30-yarder by Miller, and the keeper then fielded an overhead kick from Chris Killen after a Willo Flood cross.
Wheater almost turned another deflected Healy cross into his own net before Town had a great chance of a second just before the break when Gareth McAuley firmly met Grant Leadbitter's corner but his goalbound effort deflected wide off Healy.
Miller's shot on the turn drifted onto the roof of the net after the interval for the visitors, before Coyne had to palm away a hooked effort from Murphy following Liam Rosenior's long throw.
Coyne handled a long-range shot by Rosenior well from a half-cleared Norris corner before Lee-Barrett did well to scramble a Killen shot around the post after Justin Hoyte had headed the ball into the middle.
From the corner, Middlesbrough found their equaliser when Wheater steamed in to meet O'Neil's delivery from six yards, giving Lee-Barrett no chance.
Lee-Barrett got down smartly to block Hoyte's drive after he had been fed by Killen and then Delaney had to clear off the line when Killen's header from Hoyte's cross beat a wrong-footed Lee-Barrett.
Leadbitter's fierce drive was blocked amidst Town appeals for a penalty which ref Keith Hill waved away before Boro debutant Stephen McManus completely missed his kick when he had a chance from an O'Neil corner.
Both sides made changes as they tried to find a winning goal, and one of the replacements came closest to doing so with Middlesbrough winger Jonathan Franks' cross squirming through the legs of Lee-Barrett, but he responded quickly to grab it just in front of the line.